  The purpose of this paper is to describe geometrically discrete Lagrangian
and Hamiltonian Mechanics on Lie groupoids. From a variational principle we
derive the discrete Euler-Lagrange equations and we introduce a symplectic
2-section, which is preserved by the Lagrange evolution operator. In terms of
the discrete Legendre transformations we define the Hamiltonian evolution
operator which is a symplectic map with respect to the canonical symplectic
2-section on the prolongation of the dual of the Lie algebroid of the given
groupoid. The equations we get include as particular cases the classical
discrete Euler-Lagrange equations, the discrete Euler-Poincar\'e and discrete
Lagrange-Poincar\'e equations. Our results can be important for the
construction of geometric integrators for continuous Lagrangian systems.
