Conodonts were mostly small, elongate, eel-shaped marine animals that inhabited a variety of environments in Paleozoic and Triassic seas. Although long enigmatic, conodonts are now regarded as vertebrates and their closely controlled fossil record is not only the most extensive of all vertebrates, but it also makes conodonts the fossils of choice in upper Cambrian through Triassic biostratigraphy. Conodonts were soft-bodied except for a variety of phosphatic elements that formed a distinctive feeding apparatus. Post-mortal dissociation of the apparatus and subsequent jumbling of its elements on the sea floor led, from 1856 to about 1966, to development of an artificial, form-based taxonomy that was utilitarian, but clearly unsatisfactory as a vehicle for understanding the group in biologic terms. Natural assemblages of elements, discovered between 1879 and 1952, have been interpreted as undisturbed skeletal apparatuses, and in the mid-1960s it was determined that original composition of the apparatuses of many species could be reconstructed and statistically evaluated from collections of disjunct elements by various grouping procedures. These determinations led to an emphasis on multielement taxonomy by most (but not all) students of conodonts. Even so, only about a third of the approximately 550 valid conodont genera, have been established (or re-interpreted) in multielement terms and this makes any of the several extant schemes of suprageneric classification phylogenetically suspect. We comment on a recent scheme that recognizes 41 families assigned to some 7 orders, and suggest how it might be modified so as to square with principles of phylogenetic systematics.