Variational integrators are a class of discretizations for mechanical systems which are derived by discretizing Hamilton's principle of stationary action. They are applicable to both ordinary and partial differential equations, and to both conservative and forced problems. In the absence of forcing they conserve (multi-)symplectic structures, momenta arising from symmetries, and energy up to a bounded error. In this thesis the basic theory of discrete variational mechanics for ordinary differential equations is developed in depth, and is used as the basis for constructing variational integrators and analyzing their numerical properties. This is then taken as the starting point for the development of a new class of asynchronous time stepping methods for solid mechanics, known as Asynchronous Variational Integrators (AVIs). These explicit methods time step different elements in a finite element mesh with fully independent and decoupled time steps, allowing the simulation to proceed locally at the fastest rate allowed by local stability restrictions. Numerical examples of AVIs are provided, demonstrating the excellent properties they posess by virtue of their variational derivation.