In this talk, I will begin by providing some background on two fairly recent developments in computational mathematics: (1) the development of "variational integrators" for problems in Lagrangian mechanics; and (2) Discrete Exterior Calculus (DEC), a geometric framework for discretizing differential forms and operators, based on the cochains of algebraic topology (and closely related to mixed finite elements). After this brief review, I will discuss how these two frameworks can be combined to create structure-preserving numerical integrators for the PDEs of Lagrangian field theories. In particular, I discuss how we have done this for computational electromagnetics to construct new integrators, as well as to provide new theoretical insight into some existing methods. Finally, I will discuss some ongoing work on applying these ideas to discretize (pseudo)riemannian geometry, and the implications for numerical general relativity.