(1) The rod-shaped skeletal virgalia of lower Palaeozoic somasteroids are homologues of the major skeletal elements of modern asteroids and ophiuroids. (2) The major skeletal elements of asteroids differentiate along growth gradients which show progressive alterations in strength and direction, such that the various families can be arranged in a sequence. Soft structures differentiate in a similar manner, and yield a similar sequence. Fossil evidence indicates the older end of the sequence, which can accordingly be oriented chronologically. (3) On this basis, the Luidiidae prove to be the most archaic surviving asteroids; they are referable to the order Platyasterida, hitherto known only from lower and middle Palaeozoic fossils. Thus the soft-part characters of the Platyasterida can now be determined, and the relationship of the order to other orders of asteroids is clarified. (4) Growth gradient patterns show that Luidiidae probably arose from somasteroids, by a series of changes which can be specified. The extant asterozoan Platasterias occupies an early position in the inferred sequence, and detailed study confirms its archaic structure, showing it to be a somasteroid, though one clearly related to luidiid asteroids. (5) It is established that the earliest asteroids had: (a) Dominant transverse growth gradients in the arm, the entry of dominant longitudinal growth gradients being a post-platanasterid change. (b) No anus, and no intestine, these structures being of post-luidiid origin, at the astropectinid grade of evolution. (c) Non-suctorial tube-feet, the development of suckers being a post-astropectinid change. (d) Small, double ampullae, inherited from a pre-asteroid ancestry (identified with platasteriid somasteroids). Subsequent enlargement of the ampullae coincided with the permanent invagination of the ambulacral furrow, which occurred at the platyasterid grade of evolution, with simultaneous enlargement of the tube-feet. The single ampullae of Porcellanasteridae are thus a specialized simplification of formerly double ampullae and, being of post-luidiid origin, cannot be primitive. (e) A marginal madreporite (i.e. at the edge of the disk). (6) The mode of origin of ophiuroids from lower Palaeozoic somasteroids is deduced by methods similar to those used for finding the origin of asteroids. These predict the existence of ophiuroids with pinnate skeletal structure. Dissection confirms this, and both living and extinct ophiuroids are found to have skeletal elements homologous with virgalia. Examples are illustrated. (7) Pinnate skeletal structure is shown to be fundamental in all asterozoans, since it occurs in all somasteroids, and has been inherited from somasteroids by both asteroids and ophiuroids. (8) When the soft structures of ophiuroids are correlated with skeletal structures, the following inferences emerge: (a) Gastric caeca must originally have extended into the arm in ophiuroids, because this is true of the surviving somasteroid Platasterias, and is also true of asteroids. (b) The gonads must have originally been confined to the arms, arranged in approximately paired sequence along either side of the dorsal coelomic canal, because this condition is found to be primitive in other asterozoans. Since these conditions are fulfilled in the extant genus Ophiocanops, this ophiuroid came under suspicion as an archaic form. Dissection of its endoskeleton confirmed its archaic character, for it agrees in all major features with the zeugophiurine Oegophiurida, hitherto thought to have become extinct in the upper Carboniferous. Ophiocanops is accordingly recognized as a surviving member of the Oegophiurida, and its distinctive soft-part characters may now be attributed to that order. There are no bursae, no genital plates, no dorsal or ventral arm-plates, no oral shields, no radial shields, and the madreporite is marginal. (9) The foregoing data show which characters of asteroids and ophiuroids are ancient, and which are late, secondary features evolved after the differentiation of these two subclasses from their common somasteroid ancestry. The two categories of characters are listed. (10) Of the characters found to be ancient, all are shared with pinnulate Pelmatozoa, and have no near parallel in any other group of echinoderms. The embryonic structure of the asterozoan disk closely matches that of the crinoid calyx; it includes a centrodorsal plate, five basals, five radials, and sometimes also five infra-basals. (11) Somasteroids show transitional characters between other asterozoans and some pinnulate pelmatozoan stock, which must have resembled biserial crinoids. The somasteroid virgalia are considered to be homologues of crinoid pinnular ossicles, both being essentially rod-shaped elements, produced in similar pinnate growth gradients, and bearing a double row of cover-plates on the adoral surface. But whereas the cover-plates of crinoids border a food-groove placed on the pinnule itself, in the somasteroids the cover-plates are normally folded outwards, so as to guard a food-groove between the pinnule-like structures (metapinnules), between which an interpinnular integument has also developed. In the extant somasteroid Platasterias the cover-plates are erectile, and in the erect condition closely simulate those of crinoids, forming a double series along the adoral surface of each metapinnule. (12) Since somasteroids are themselves antecedent to other asterozoans, it follows that all star-shaped echinoderms (Asterozoa) constitute a single phylogenetic stock, unrelated to any other echinoderms save pinnulate pelmatozoans. (13) Embryological data suggest that asterozoans evolved from a pentacrinoid stage of development of some pinnulate pelmatozoan, as a consequence of a dislocation of the main radial growth gradient at the junction of the radial calycinal plate and the first brachial ossicle. The dislocation was such that the first brachial ossicles were deflected adorally. Comparative anatomy shows that subsequent dislocations of the transverse (i.e. pinnular) growth gradients caused the second virgalium in each metapinnule to be ejected from the growth axis to which it belonged, to become the occluded superambulacral ossicle; in post-astropectinid asteroids this occluded ossicle usually is aborted, and it is unknown at present in any ophiuroid. In the final phases of evolution in both asteroids and ophiuroids the dislocations of transverse axes are so severe as to result in their complete disintegration, the ossicles now falling under the control of dominant longitudinal gradients. (14) Crinoids are the only Pelmatozoa which show structure comparable to that of Asterozoa. Their characters can be segregated into two groups, namely (a) those which are shared with somasteroids, and (b) those which are shared with other, more generalized and archaic pelmatozoans, such as eocrinoids or cystoids. Crinoids can therefore be considered as late pelmatozoans which have: (i) retained the original meridional patterns of growth in the calyx, as an inheritance from earlier pelmatozoans, and (ii) acquired in addition five radially directed, divergent axes of growth in the horizontal plane, leading to the outgrowth of arms and, later on, to the secondary acquisition of pinnules. Somasteroids would then be a further term in this progression, in which the first character is diminished, and the second exploited. All Asterozoa would therefore be derived (through somasteroids) from crinoids, or some crinoid-like group of Pelmatozoa. (15) In sharp contrast, the growth gradients of the echinozoan echinoderms (i.e. Echinoidea, Holothuroidea, and probably also Ophiocistiodea) are comparable only with those of ancient, pre-crinoid pelmatozoans. All these have either meridional, or quite disordered, growth gradients. None has radial divergent components. The Echinozoa are accordingly viewed as archaic derivatives of a still more ancient globoid pelmatozoan stock. (16) The conventional association of the Echinozoa with the Asterozoa, under one head, namely the subphylum Eleutherozoa, is now unacceptable. The characters attributed to the Eleutherozoa are in reality shared only by demonstrably late groups of Asterozoa, and demonstrably early groups of Echinozoa. The subphylum Eleutherozoa is polyphyletic, and must be abandoned as a formal unit of classification. Instead, two subphyla, Asterozoa and Echinozoa, are indicated. (17) Auricularia and pluteus larval stages of those echinoderms known to possess them cannot reflect broad phylogenetic relationships, because they imply a closer relationship between ophiuroids and echinoids, than between ophiuroids and asteroids, and this is demonstrably false. Such larvae must therefore have arisen as secondary phases of development, presumably as responses to the demands of planktonic life during temporary food-gathering stages of development.