The symbol '$' below indicates a synapomorphy, (a character believed to be uniquely present in the last common ancestor of all heterodontiforms).
The heterodontiforms are morphologically rather distinctive. The mouth is located entirely anterior to the orbits. Labial cartilages are found in the most anterior part of the mouth. Nasoral grooves are present, connecting the external nares to the mouth. The nasal capsules are "trumpet-shaped" & well-separated from orbits. $ Circumnarial skin folds present; but $ the rostral process of neurocranium (braincase) absent, although a precerebral fossa is present. Finally, the braincase bears a supraorbital crest.
The eyes lack a nictitating fold. A spiracle is present, but small. The dorsal ends of branchial arches 4 & 5 are attached, but not fused into a "pickaxe" as in lamniform sharks. Heterodontiforms have 2 dorsal fins, with fin spines, as well as an anal fin. The dorsal and anal fin also contain basal cartilages -- not just fin rays.
The Heterodontiforms appear in the fossil record in the Early Jurassic, well before any of the other Galeomorphii, a group which includes all modern sharks except the dogfish and its relatives. However, they have never been common, and it is likely their origin lies even further back.