A fish is any member of a paraphyletic group of organisms that consist of all gill-bearing aquatic craniate animals that lack limbs with digits.Included in this definition are the living hagfish, lampreys, and cartilaginous and bony fish, as well as various extinct related groups.ish do not represent a monophyletic group, and therefore the evolution of fish is not studied as a single event. Early fish from the fossil record are represented by a group of small, jawless, armored fish known as Ostracoderms. Jawless fish lineages are mostly extinct.