We first define spaces of half-integral weight modular forms of general p-adic weight equipped with a Hecke action, generalizing earlier work in which the weight was required to be classical. We then define Banach modules of families of half-integral weight modular forms equipped with a Hecke action. In both cases the Hecke operators are proven to be continuous and Up2 is proven to be compact in the overconvergent case. These Banach modules are then used to constuct an eigencurve parameterizing systems of eigenvalues of the Hecke operators acting on the spaces of half-integral weight modular forms. Along the way a Coleman-style theorem indicating that eigenforms of low slope are classical is proven.