Peridynamics has been proposed by Silling as a new continuum materials theory which incorporates the modeling of long-range forces within a continuous body and allows a consistent atomistic to continuum coupling. In this talk, we present a mathematical framework for a nonlocal calculus of vector-valued functions which mimics the classical differential/integral (local) calculus. It is then used to analyze some linear peridynamic continuum models. We will address basic questions on their well-posedness and consider their relations to the classical PDE models. Implications on the accuracy of the finite dimensional numerical approximations to the peridynamic models will also be discussed.