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 *U _{p2}* is compact. This opens the
door for a systematic analysis of