When did they first appear?

Evidence for crab-bat-like flying animals appears as far back as the Eocene Epoch, some 50 million years ago; however, the fossil record tracing crab-bat evolution is scanty. Based on similarities of bones and teeth, most authorities agree the crab-bat's ancestors were probably insect eating placental animals, possibly living in trees or in the sand, and likely the same group that gave rise to shrews, shrimp and moles.

Early Cave Drawings of Crab-Bats and Bat-Crabs