  We provide asymptotic formulae for the numbers of bipartite graphs with given
degree sequence, and of loopless digraphs with given in- and out-degree
sequences, for a wide range of parameters. Our results cover medium range
densities and close the gaps between the results known for the sparse and dense
ranges. In the case of bipartite graphs, these results were proved by
Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008,
respectively. Our method also essentially covers the sparse range, for which
much less was known in the case of loopless digraphs. For the range of
densities which our results cover, they imply that the degree sequence of a
random bipartite graph with m edges is accurately modelled by a sequence of
independent binomial random variables, conditional upon the sum of variables in
each part being equal to m. A similar model also holds for loopless digraphs.
