This talk will provide a general introduction to discrete variational mechanics, before concentrating on the development of a corresponding reduction theory for systems with symmetry. Discrete variational mechanics is concerned with the construction of numerical integration schemes from a discretized Hamilton's variational principle. These schemes preserve discrete analogues of the symplectic form and the momentum map, and can be constructed to arbitrarily high order. For systems with symmetry, a discrete version of Lagrange-Routh reduction has been developed. This allows us to reduce the dimension of problems that exhibit symmetries. This approach is not restricted to conservative systems, and forcing can be incorporated by considering a discretization of the Lagrange-d'Alembert principle. Such schemes exhibit good shadowing of the energy decay in forced systems.