In this paper we obtain a combinatorial lower bound δ g ( G) for the crossing number cr g ( G) of a graph G in the closed orientable surface of genus g, and we conjecture that equality holds in a wide range of interesting cases. The lower bound is applied to the crossing number of the 1-skeleton of a d-dimensional cube to show that this crossing number must be at least 4, and a constructive technique is used to show that the crossing number is at most 8. Finally, we show that the crossing number of any graph is at most k 2 times the crossing number of the underlying simple graph, where k = maximum multiplicity of an edge.