  We investigate the approximation of Monge--Kantorovich problems on general
compact metric spaces, showing that optimal values, plans and maps can be
effectively approximated via a fully discrete method. First we approximate
optimal values and plans by solving finite dimensional discretizations of the
corresponding Kantorovich problem. Then we approximate optimal maps by means of
the usual barycentric projection or by an analogous procedure available in
general spaces without a linear structure. We prove the convergence of all
these approximants in full generality and show that our convergence results are
sharp.
