The first Archaeopteryx was found in 1861 near Solnhofen in Bavaria, Germany. Archaeopteryx looks like a reptile with wings and feathers. It had a mouth with teeth, claws at the hands and a long tail like dinosaurs or reptiles. Today, it is known that some dinosaurs looked like birds and that some had feathers. When they are born, today's South American hoatzin have claws on their wings when they are young, just like Archaeopteryx.
Archaeopteryx lived in the early Jurassic Period, 150.8–148.5 million years ago. Most of the specimens of Archaeopteryx that have been found come from the Solnhofen limestone in Bavaria, Germany, which is a lagerstätte, a rare and stunning geological formation known for its superbly full fossils. 
Archaeopteryx was close to the size of a raven, with broad wings that were rounded at the ends and a long tail. It could reach up to 20 inches in length, with a weight of 1.8 to 2.2 pounds. Archaeopteryx feathers, though less documented than its other traits, looked a lot like bird feathers in structure and design. But,Archaeopteryx had a lot of theropod dinosaur traits too. Unlike modern birds, Archaeopteryx had small teeth  as well as a long bony tail, traits which Archaeopteryx shared with other dinosaurs of the time.
As it shows quite a few traits of both birds and dinosaurs, Archaeopteryx has often been deemed a link between them. In the 1970s, John Ostrom, following T. H. Huxley's lead in 1868, argued that birds evolved within theropod dinosaurs and Archaeopteryx was a critical piece of evidence for this argument; it had a number of bird traits, such as a wishbone, flight feathers, wings and a partially reversed first toe, and a number of dinosaur and theropod features. For instance, it has a long ascending process of the ankle bone, interdental plate, an obturator process of the ischium, and long chevrons in the tail. In particular, Ostrom found that Archaeopteryx was remarkably similar to the theropod family Dromaeosauridae.        
The first bones of Archaeopteryx were found in 1861; just two years after Charles Darwin wrote On the Origin of Species. Archaeopteryx seemed to prove Darwin's theories and has since been a key piece of proof for the origin of birds, the transitional fossil debate, and proof of evolution. Indeed, more research on dinosaurs from the Gobi Desert and China has since given more proof of a link between Archaeopteryx and the dinosaurs, such as the Chinese feathered dinosaurs. Archaeopteryx is close to the ancestry of modern birds, and it shows most of the features one would expect in an ancestral bird. But, it may not be the direct ancestor of living birds, and it is not clear how much evolutionary divergence was already found in other birds at the time.
To this day, most of the fossils are classed in a single species A. lithographica, but the taxonomic history is complex. A lot of names have been put out for the specimens, most of which are just spelling errors. As interpreted today, the name A. lithographica just referred to the single feather described by Christian Erich Hermann von Meyer. But in 1954, Gavin de Beer said that the London specimen was the holotype. So in 1960, Swinton proposed that the name Archaeopteryx lithographica be placed on the official genera list making the alternative names Griphosaurus and Griphornis invalid.  The ICZN, fully accepting de Beer's view, did indeed hold back the alternative names at first proposed for the first skeleton specimens,   In 1977 the first specific name of the Haarlem specimen, crassipes, described by von Meyer as a pterosaur before its true nature was known, was also held back.  
The relationships of the specimens are hard. Most later specimens now have their own species. The Berlin specimen has been classed as Archaeornis siemensii, the Eichstätt specimen as Jurapteryx recurva, the Munich specimen as Archaeopteryx bavarica and the Solnhofen specimen was classed as Wellnhoferia grandis.
Recently, it has been claimed that all the specimens are from the same species. But, there are big differences in the specimens. In particular, the Munich, Eichstätt, Solnhofen and Thermopolis specimens differ from the London, Berlin, and Haarlem specimens in being smaller or much larger, having different finger proportions, having more slender snouts, lined with forward-pointing teeth and likely presence of a sternum. These differences are as large as or larger than the differences seen today between adults of different bird species. But, it could be that these differences could be explained by different ages of the living birds.
Last, some note that the feather, the first specimen of Archaeopteryx described, does not agree well with the flight-related feathers of Archaeopteryx. It's clearly a flight feather of a species of the time, but its size is a sign that it may be from to another, smaller species of feathered theropod, of which just this feather is so far known.  As the feather was in the early 21st century seen as the type specimen, this would have made significant nomenclatorial confusion because the name Archaeopteryx should then no longer be applied to the skeletons. In 2007, two sets of scientists thus petitioned the ICZN asking that the London specimen be clearly made the type by classing it as the new holotype specimen. This was upheld by the ICZN after four years of debate, and the London specimen was classed on October 3, 2011.
If two names are given, the first means the original describer of the "species", the second the author on whom the given name combination is based. As always in zoological nomenclature, putting an author's name in parentheses means that the taxon was first described in a different genus.
- Pterodactylus crassipes Meyer, 1857 [suppressed in favor of A. lithographica 1977 per ICZN Opinion 1070]
- Rhamphorhynchus crassipes (Meyer, 1857) (as Pterodactylus (Rhamphorhynchus) crassipes) [suppressed in favor of A. lithographica 1977 per ICZN Opinion 1070]
- Archaeopteryx lithographica Meyer, 1861 [nomen conservandum]
- Scaphognathus crassipes (Meyer, 1857) Wagner, 1861 [suppressed in favor of A. lithographica 1977 per ICZN Opinion 1070]
- Archaeopterix lithographica Anon., 1861 [lapsus]
- Griphosaurus problematicus Wagner, 1862 [nomen oblitum 1961 per ICZN Opinion 607]
- Griphornis longicaudatus Woodward, 1862 [nomen oblitum 1961 per ICZN Opinion 607]
- Griphosaurus longicaudatum (Woodward, 1862) [lapsus]
- Griphosaurus longicaudatus (Owen, 1862) [nomen oblitum 1961 per ICZN Opinion 607]
- Archaeopteryx macrura Owen, 1862 [nomen oblitum 1961 per ICZN Opinion 607]
- Archaeopterix macrura Owen, 1862 [lapsus]
- Archaeopterix macrurus Egerton, 1862 [lapsus]
- Archeopteryx macrurus Owen, 1863 [unjustified emendation]
- Archaeopteryx macroura Vogt, 1879 [lapsus]
- Archaeopteryx siemensii Dames, 1897
- Archaeopteryx siemensi Dames, 1897 [lapsus]
- Archaeornis siemensii (Dames, 1897) Petronievics, 1917
- Archaeopteryx oweni Petronievics, 1917 [nomen oblitum 1961 per ICZN Opinion 607]
- Gryphornis longicaudatus Lambrecht, 1933 [lapsus]
- Gryphosaurus problematicus Lambrecht, 1933 [lapsus]
- Archaeopteryx macrourus Owen, 1862 fide Lambrecht, 1933 [lapsus]
- Archaeornis siemensi (Dames, 1897) fide Lambrecht, 1933? [lapsus]
- Archeopteryx macrura Ostrom, 1970 [lapsus]
- Archaeopteryx crassipes (Meyer, 1857) Ostrom, 1972 [suppressed in favor of A. lithographica 1977 per ICZN Opinion 1070]
- Archaeopterix lithographica di Gregorio, 1984 [lapsus]
- Archaeopteryx recurva Howgate, 1984
- Jurapteryx recurva (Howgate, 1984) Howgate, 1985
- Archaeopteryx bavarica Wellnhofer, 1993
- Wellnhoferia grandis Elżanowski, 2001
The last four taxa may be valid genera and species.
"Archaeopteryx" vicensensis (Anon. fide Lambrecht, 1933) is a nomen nudum for what seems to be an undescribed pterosaur.
Through the years, 11 body fossil specimens of Archaeopteryx and a feather that may come from it have been found. All of the fossils come from the limestone beds, quarried for centuries, near Solnhofen, Germany.  
The first finding, one feather, was dug up in 1860 or 1861 and described in 1861 by Christian Erich Hermann von Meyer. It is now at the Humboldt Museum für Naturkunde in Berlin. This is generally assigned to Archaeopteryx and was the first holotype, but if it's a feather of this species or some other proto-bird is not known. There are some signs it is in fact not from the same animal as most of the skeletons (the "typical" A. lithographica). 
The first skeleton, known as the London Specimen (BMNH 37001),  was dug up in 1861 near Langenaltheim, Germany and perhaps given to a local physician Karl Häberlein in return for medical services. He then sold it to the Natural History Museum in London.  Missing most of its head and neck, it was described in 1863 by Richard Owen as Archaeopteryx macrura, allowing for the chance it did not come from to the same species as the feather. In the later 4th edition of his On the Origin of Species,  Charles Darwin wrote how some authors had claimed "that the whole class of birds came suddenly into existence during the eocene period; but now we know, on the authority of Professor Owen, that a bird certainly lived during the deposition of the upper greensand; and still more recently, that strange bird, the Archeopteryx, with a long lizard-like tail, bearing a pair of feathers on each joint, and with its wings furnished with two free claws, has been discovered in the oolitic slates of Solnhofen. Hardly any recent discovery shows more forcibly than this how little we as yet know of the former inhabitants of the world."
The Greek term "pteryx" (πτέρυξ) means "wing", but can also just mean "feather". Von Meyer suggested this in his description. At first he referred to one feather which looked like a bird's wing feather, but he had heard of and been shown a rough sketch of the London specimen, to which he referred as a "Skelet eines mit ähnlichen Federn bedeckten" ("skeleton of an animal covered in similar feathers"). In German, this vagueness is solved by the term Schwinge which does not necessarily mean a wing used for flying. Urschwinge was the choice translation of Archaeopteryx among German scholars in the late 19th century. In English, "ancient pinion" gives a rough guess.
Since then nine specimens have been recovered:
The Berlin Specimen (HMN 1880) was found in 1874 or 1875 on the Blumenberg near Eichstätt, Germany, by farmer Jakob Niemeyer. He sold this fossil in 1876, to inn-keeper Johann Dörr, who sold it to Ernst Otto Häberlein, the son of K. Häberlein. Placed on sale from 1877 and 1881, with likely buyers such as O.C. Marsh of Yale University's Peabody Museum, it was bought by the Humboldt Museum für Naturkunde, where it is now displayed. The deal was payed for by Ernst Werner von Siemens, founder of the famous company that bears his name.  Described in 1884 by Wilhelm Dames, it is the most complete specimen, and the first with a full head. It was in 1897 named by Dames as a new species, A. siemensii; a recent test backs up the A. siemensii species identification. 
Made up of a chest, the Maxberg Specimen (S5) was found in 1956 near Langenaltheim; it was brought to the attention of professor Florian Heller in 1958 and described by him in 1959. It is now missing, though it was once at the Maxberg Museum in Solnhofen. It was owned by Eduard Opitsch, who loaned it to the museum till 1974. When he died in 1991, the specimen was discovered to be missing and may have been stolen or sold. The specimen is missing its head and tail, though the rest of the skeleton is mostly in one piece.
The Haarlem Specimen (TM 6428, aka the Teyler Specimen) was found in 1855 near Riedenburg, Germany and described as a Pterodactylus crassipes in 1877 by von Meyer. It was reclassified in 1970 by John Ostrom and is now at the Teylers Museum in Haarlem, the Netherlands. It was the very first specimen, in spite of the classification error, and it is one of the least complete specimens, mostly made up of limb bones and isolated neck vertebrae and ribs.
The Eichstätt Specimen (JM 2257) was found in 1951 near Workerszell, Germany and described by Peter Wellnhofer in 1974. Now at the Jura Museum in Eichstätt, Germany, it is the smallest specimen and has the 2nd best head. It may be a separate genus (Jurapteryx recurva) or species (A. recurva).
The Solnhofen Specimen (BSP 1999) was found in the 1970s near Eichstätt, Germany and described in 1988 by Wellnhofer. Now at the Bürgermeister-Müller-Museum in Solnhofen, it was first classified as Compsognathus by an amateur collector, the same burgomaster Friedrich Müller after which the museum is named. It is the largest specimen known and may come from a separate genus and species, Wellnhoferia grandis. It is just missing parts of the neck, tail, backbone, and head.
The Munich Specimen (S6, formerly known as the Solnhofen-Aktien-Verein Specimen) was found on August 3rd, 1992 near Langenaltheim and described in 1993 by Wellnhofer. It is now at the Paläontologisches Museum München in Munich, to which it was sold in 1999. What was first thought to be a bony sternum turned out to be part of the coracoid.  Just the front of its face is missing. It may be a new species, A. bavarica.
An 8th, incomplete specimen was found in 1990, not in Solnhofen limestone, but in somewhat younger sediments at Daiting, Suevia. So it's known as the Daiting Specimen, and had been known since 1996 just from a cast, shown at the Naturkundemuseum in Bamberg for a short time. Long remaining hidden and therefore dubbed the 'Phantom', the first was bought by palaeontologist Raimund Albertsdörfer in 2009.  It was on display for the first time with 6 more original fossils of Archaeopteryx at the Munich Mineral Show in October 2009.  A quick look by scientists shows that this specimen might be a new species of Archaeopteryx. It was found in a limestone bed that was a few hundred thousand years younger than most finds. 
Another fragmentary fossil was found in 2000. It is in private hands and since 2004 on loan to the Bürgermeister-Müller Museum in Solenhofen, so it is called the Bürgermeister-Müller Specimen; the institute itself officially refers to it as the "Exemplar of the families Ottman & Steil, Solnhofen". As the piece represents the remains of one wing of Archaeopteryx, the popular name of this fossil is "chicken wing".
Long in a private collection in Switzerland, the Thermopolis Specimen (WDC CSG 100) was found in Bavaria and described in 2005 by Mayr, Pohl, and Peters. Sent to the Wyoming Dinosaur Center in Thermopolis, Wyoming, it has the best-preserved head and feet; most of the neck and the lower jaw have not been preserved. The "Thermopolis" specimen was described in the December 2, 2005 Science journal article as "A well-preserved Archaeopteryx specimen with theropod features"; it shows that the Archaeopteryx lacked a reversed toe — a universal feature of birds — limiting its ability to perch on branches and implying a ground-based or trunk-climbing lifestyle.  This may be proof of theropod ancestry. In 1988, Gregory S. Paul claimed to have found proof of a hyperextensible second toe, but this was not proved and accepted by other scientists till the Thermopolis specimen was described.  "Until now, the feature was thought to belong only to the species' close relatives, the deinonychosaurs." 
The Thermopolis Specimen was assigned to Archaeopteryx siemensii in 2007. Now on loan to the Royal Tyrrell Museum, Drumheller, Alberta, Canada, it's deemed the most complete and well preserved Archaeopteryx yet. 
In 2011 an 11th specimen was found. It is said to be one of the more complete specimens, but is missing the skull and one forelimb. It is privately owned and has yet to be named or described scientifically.  
Specimens of Archaeopteryx were best known for their well-developed flight feathers. They were crooked and showed the form of flight feathers in birds, with stable vanes.  The tail feathers were less crooked and had firm vanes, just like in birds, but the thumb did not yet bear a separately movable tuft of stiff feathers.
The body plumage of Archaeopteryx is less well documented and has just been properly researched in the well-preserved Berlin specimen. Thus, as more than one species seems to be involved, the research of the Berlin specimen's feathers does not necessarily hold true for the rest of the species of Archaeopteryx. In the Berlin specimen, there are "trousers" of well-developed feathers on the legs; some of which seem to have a plain form feather structure but are quite decomposed.But, in part they are firm and thus can support flight. 
A patch of feathers ran down the back which looked much like those of birds in that they were aligned and firm, though not as stiff as the flight-related feathers. Apart from that, the feather traces in the Berlin specimen are limited to a sort of "proto-down" like that found in the dinosaur Sinosauropteryx, decomposed and soft, and might have looked more like fur than like feathers in life (though not in their microscopic structure). These are on the rest of the body, as far as such structures are both preserved and not wiped out by preparation, and the lower neck. 
But, there is no sign of feathers on the upper neck and head. While these may have been nude, this may still be an artifact of preservation. It seems that most Archaeopteryx specimens became embedded in anoxic sediment after drifting some time on their back in the sea—the head and neck and the tail are for the most part bent down, which may mean that the specimens had just started to rot when they were embedded, with tendons and muscle relaxing so that the characteristic shape of the fossil specimens was gained. This would mean that the skin was already softened and loose, which is helped by the fact that in some specimens the flight feathers were starting to come off at the point of embedding in the sediment. So it is thought that the pertinent specimens moved on the edge the sea bed in shallow water for some time till buried, the head and upper neck feathers sloughed off, while the more firmly attached tail feathers remained. 
In 2011, graduate student Ryan Carney and colleagues did the first colour study on an Archaeopteryx specimen. The team found the structure of melanosome in the single-feather specimen described in 1861. The resultant structure was then compared to that of 87 bird species and was shown to very likely be black. The feather studied was most likely one covert, which would have covered part of the main feathers on the wings. While the study does not mean that Archaeopteryx was all black, it does hint that it had some black color such as the coverts. Carney pointed out that this is in line with what we know of modern flight traits, in that black melanosomes have structural properties that make feathers strong for flight.
- All About Archaeopteryx from Talk.Origins.
- Use of SSRL X-ray takes 'transformative glimpse' – A look at chemicals linking birds and dinosaurs.
- Archaeopteryx: An Early Bird – University of California Museum of Paleontology.
- Are Birds Really Dinosaurs? – University of California Museum of Paleontology.
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