A space of matrices of constant rank is a vector subspace V, say of dimension n+1, of the set of matrices of size axb over a field k, such that any nonzero element of V has fixed rank r. It is a classical problem to look for examples of such spaces of matrices, and to give relations among the possible values of the parameters a,b,r,n. In this talk I will report on my latest joint projects (and work in progress) with D. Faenzi, D. Fratila and P. Lella, where we use techniques involving vector bundles on projective spaces, finitely generated graded modules, and representation theory.