Class of fleshy-finned fish comprising two main groups (superorders): the coelacanths (Crossopterygii), and the lungfish (Dipnoi). The Crossopterygii appeared in the Devonian, were widely distributed in the Mesozoic, and are now represented by a single genus, Latimeria. One order of the Crossopterygii, the Rhipidistia, is credited with being the root stock for more advanced vertebrates. The Dipnoi also appeared in the Devonian and are now represented by three species. They inhabit fresh water and have developed organs by which they can breathe atmospheric air.