Consider an $n$ by $n$ random matrix $H$ with independent entries. As the dimension grows to infinity, how likely is $H$ to be invertible? And what is the typical norm of the inverse? These questions can be traced back to P. Erdos (for matrices with +1,-1 entries) and von Neumann and his collaborators (motivated by the analysis of numerical algorithms). For both matrices with all independent entries and for symmetric random matrices, there was a considerable progress on the invertibility problem in the last few years. The methods come from different areas, including classical random matrix theory, mathematical physics, geometrical functional analysis, and additive combinatorics. A related problem for rectangular random matrices is motivated by statistical applications (covariance estimation). We will discuss recent progress and several conjectures.