The purpose of this paper is to examine the evidence afforded by the chimaeroids and fossil holocephalans on the origin and evolution of the Holocephali, and, in particular, on the hypothesis recently proposed by Ørvig of a relationship between chimaeroids and ptyctodont arthrodires. Members of all the fossil holocephalan groups which are known by moderately complete specimens except the petalodonts and edestids are described, particular attention being paid to the histology of the dermal skeleton. Three species from the English Lower Lias,Squaloraja polyspondylaAgassiz,Myriacanthus paradoxusAgassiz andMetopacanthus granulatus(Agassiz) (a species previously placed inMyriacanthus) are described in detail. From the British Lower CarboniferousDeltoptychius armigerus(Traquair), previously placed inOracanthus, is described in detail and two new species,Deltoptychius moythomasisp.nov. andMenaspacanthus armaghensisgen. and sp.nov., are described on fragmentary material. The genusOracanthusAgassiz is reviewed: it is shown that the type species is probably acanthodian but that many species have been wrongly included in the genus. There is found to be strong evidence that the chimaeroids are derived, through the Jurassic myriacanthids and allied forms, from a Palaeozoic group which includes the PermianMenaspisand whose most primitive known genus isDeltoptychius. The most striking feature ofDeltoptychiusis the dermal armour of dentine-like tissue on the head which includes a complete head shield bearing a pair of sensory canals. This armour is of a peculiar type which is not homologous with the armour of the arthrodires, and appears to be confined to the Chimaeriformes. It is suggested thatHelodusis not directly related to the Chimaeriformes. These conclusions are embodied in a new classification of the Holocephali in whichHelodusis made the type of a new order Helodontiformes and in which the Chimaeriformes is divided into four sub-orders, Chimaeroidei, Squalorajoidei, Myriacanthoidei and Menaspoidei, with new families being made in the last two sub-orders. On the basis of the information obtained from the systematic part, the origin and evolution of the various characters of chimaeroids are discussed. The ethmoid canal appears to be a recent acquisition which occurs only in Chimaeroidei. The tritors on chimaeroid tooth plates are another recent adaptation. There is no convincing evidence that the teeth of Holocephali were primitively numerous, as they are inHelodus, petalodonts and edestids: this ‘selachian’ type of dentition is quite possibly secondarily derived from a 1chimaeroid ’ type of dentition. The dorsal fin spine of the chimaeroids appears to be a recent adaptation which first appears in the myriacanthoids and has arisen independently, being unrelated to the fin spine ofHelodus. A series of changes has taken place in the structure of the fin spine in Chimaeriformes which is similar to the changes which have occurred in the fin spines of sharks. The scales ofDeltoptychiusare cyclomorial: the change from a cyclomorial to a placoid (synchronomorial) scale has taken place independently among elasmobranchs at least three times: in the selachians, in the edestids, and in the Chimaeriformes. The scales ofHelodusare synchronomorial: this is the only Palaeozoic elasmobranch in which this condition has yet been found. The crescentic calcifications surrounding the sensory canals of chimaeroids are shown to be modified scales. The pre-pelvic tenacula of male chimaeroids have evolved from groups of enlarged scales without skeletal support which occur in myriacanthoids. It is suggested that the frontal clasper of male chimaeroids has evolved from paired structures, defensive in function, which occurred in both sexes in Palaeozoic Chimaeriformes. The dermal armour of Chimaeriformes is formed by fusion of scales. It has undergone a series of changes in histological structure which is similar to that undergone by the dorsal fin spine, but which took place much earlier in time. The evidence for a relationship between Holocephali and the selachians or arthrodires is reviewed. It is concluded that there is evidence of relationship between holocephalans and arthrodires, but that the resemblances between the two groups are in general features only. Mainly because of the structure of the exoskeleton inDeltoptychius, a relationship with the ptyctodonts is very unlikely. The holocephalans seem to be closest to the most primitive arthrodires (rhenanids and stensioellids), suggesting that the two groups share a common ancestor but are not directly related.