  We present a geometric framework for discrete classical field theories, where
fields are modeled as "morphisms" defined on a discrete grid in the base space,
and take values in a Lie groupoid. We describe the basic geometric setup and
derive the field equations from a variational principle. We also show that the
solutions of these equations are multisymplectic in the sense of Bridges and
Marsden. The groupoid framework employed here allows us to recover not only
some previously known results on discrete multisymplectic field theories, but
also to derive a number of new results, most notably a notion of discrete
Lie-Poisson equations and discrete reduction. In a final section, we establish
the connection with discrete differential geometry and gauge theories on a
lattice.
