A matrix $C$ has the Monge property if $c_{ij} + c_{IJ} \leq c_{Ij} + c_{iJ}$ for all $i < I$ and $j < J$. Monge matrices play an important role in combinatorial optimization; for example, when the transportation problem (resp., the traveling salesman problem) has a cost matrix which is Monge, then the problem can be solved in linear (resp., quadratic) time. For given matrix dimensions, we define the Monge polytope to be the set of nonnegative Monge matrices normalized with respect to the sum of the entries. In this paper, we give an explicit description and enumeration of the vertices, edges, and facets of the Monge polytope; these results are sufficient to construct the face lattice. In the special case of two-row Monge matrices, we also prove a polytope volume formula. For symmetric Monge matrices, we show that the Monge polytope is a simplex and we prove a general formula for its volume.