We introduce a formalism for studying modular forms of half-integral weight that is geometric in nature in the sense that it identifies such forms with sections of certain sheaves on modular curves. This formalism allows one to define modular forms of half-integral weight over a very general class of rings. These forms enjoy all of the familiar properties of their integral weight counterparts such as a q-expansion principle and a geometrically defined Hecke action. Finally, this formalism is used to define spaces of p-adic modular forms of (classical) half-integral weight. These spaces are equipped with a Hecke action for which the Up2 is compact. This opens the door for a systematic analysis of p-adic properties of modular forms of half-integral weight via the spectral theory of the Up2 operator.