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Confusingly, coelacanth isn't the name of a specific genus or species of fish, but rather of an ancient family that comprises dozens of genera and species. The coelacanths were lobe-finned fish that first evolved in the late Devonian period (about 360 million years ago), and were closely related to the first tetrapods, that is, the fish that evolved rudimentary legs and crawled up out of the shallow seas onto dry land (thus inaugurating hundreds of millions of years of vertebrate evolution). As a group, coelacanths were characterized by their stalked fins, tiny hearts and brains, and "rostral organs" on their heads that may have been used to sense prey electrically. (See 10 Facts About Coelacanths.)
Although they're interesting in their own right, coelacanths are famous for another reason: these fish were believed to have gone extinct 65 million years ago, at the end of the Cretaceous period, until a live specimen (of the genus Latimeria) was caught off the coast of South Africa in 1938, and another species of Latimeria in 1998 near Indonesia. These discoveries have fueled the fantasies of cryptozoologists, who believe that the discovery of live coelacanths opens the possibility for the continued survival of dinosaurs (and other supposedly long-extinct creatures) into modern times. We just haven't found them yet!