Specimens from the Carboniferous Granton shrimp bed of Edinburgh, Scotland, provide the most complete record of conodont anatomy. Ten specimens are now known, six of which are previously undescribed, and form the basis of a new description and restoration of the conodont animal. The feeding apparatus is present in eight of the specimens; all but two of these can be assigned to Clydagnathus on the basis of the elements. A different genus and species is represented by the other two. The soft tissue morphology of all the specimens is similar. The Granton animals are elongate, 21-55 mm in preserved length with a short head, a trunk with V-shaped myomeres, and a ray-supported caudal fin. The head is characterized by two lobate structures, which are interpreted as hollow sclerotic cartilages indicating the position of large eyes. One specimen preserves traces of possible otic (auditory) capsules and branchial structures. Ventral and immediately posterior to the eyes lies the feeding apparatus, with the ramiform elements at the anterior end. There is no evidence of tissue surrounding this apparatus, indicating incomplete preservation of ventral soft parts, at least at the anterior end of the specimens. The trunk of most specimens displays the notochord as a pair of axial lines which represent its margins. In one specimen the area between the two lines is mineralized and displays a transverse fibrous structure. The notochord tapers anteriorly and posteriorly; it extends as far forwards as the ramiform elements of the feeding apparatus but does not reach the anterior tip of the head. Two specimens show a possible trace of the dorsal nerve cord. V-shaped myomeres are particularly well-preserved along the trunk of some of the new specimens; their preservation as distinct chevrons is attributed to a little post-mortem shrinkage. Possible traces of original muscle fibres are preserved in the myomeres of one specimen. The tail is present on two specimens, with fin rays representing a caudal fin that may be symmetrical or may be slightly more extensive on the ventral margin. The evidence of the soft-part anatomy, together with features of element histology, show that the conodonts are vertebrates. Hypotheses that conodonts have affinities with nemerteans, molluscs, chaetognaths, or cephalochordates are refuted. Nor do the conodonts represent a separate phylum. Within the Vertebrata, the conodonts are considered to lie crownwards of the myxinoids, forming a primitive sister group of the Heterostraci + Myopterygii.