Let Q(n) denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number v(Q(n)), i.e., the minimum number of edge-crossings in any planar drawing of Q(n). The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let N = 2n be the number of vertices of Q(n). We show that v(Q(n)) < 1/6N2. For the lower bound we prove that v(Q(n)) = OMEGA(N(lg N)c lg lgN), where c > 0 is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is v(Q(n)) = OMEGA(N(lg N)2). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.