Recent claims that conodonts are members of the Craniata or Vertebrata are based in part upon soft tissue features that have been preserved in a small number of specimens. These features include what appear to be radials in the caudal fin and paired structures that have been identified as eye remnants. The evidence for radials is limited, but credible. However, the anatomy of extant cyclostomes suggests that the paired structures are more reasonably interpreted as otic capsules than the remnants of sclerotic eye capsules. Moreover, even if these structures are the remnants of eyes, conodonts might equally well be a sister group to the craniates as a member of that group. Aside from these paired structures, conodont fossils exhibit no features that are suggestive of a cartilaginous skeleton. Given that cyclostome fossils sometimes show evidence of the cartilages of the head, the apparent absence of a similar skeleton in conodont animals calls into question the claim that they are craniates. The simple single chevron shape of conodont myomeres also suggests that they lie outside of the Craniata. All living craniates have double-chevron myomeres as adults, whereas simple myomeres of the conodont type are found in the non-craniate cephalochordates. Thus the available soft tissue evidence suggests that conodonts are best regarded as the sister group of the craniates.