Most of the type species of 67 active genera and of genera considered as synonyms of them are redescribed and refigured. Three new genera are described. A new classification is proposed, including 3 new families, based on lineages suggested by several evolutionary trends within the order. Among these trends is reduction from 2 to 1 pore in each ambulacral plate beyond the petals, introduction of buccal pores, increase in the area below the petals, change from a tetrabasal to a monobasal apical system, and increase in size of the adoral tubercles. The order arose in the Early Jurassic, reached its maximum number of species in the Eocene, then abruptly declined, with only 16 species living today. It is suggested that this reduction was caused by the cooling of the seas after the Eocene.