On 0-1 matrices with prescribed row and column sums

Matrices with 0-1 entries and prescribed row and column sums is a classical object in combinatorics. In this talk, I'll present a new (and only) asymptotic formula for the number of such matrices and describe what a random 0-1 matrix with prescribed row and column sums looks like. Depending on the wishes of the audience, I can either sketch a proof or describe what changes if instead of 0-1 matrices we consider non-negative integer matrices.