We discuss some recent progress on the Monotone Column Permanent (MCP) conjecture. We use a general method for proving that a univariate polynomial has real roots only, namely by showing that a corresponding multivariate polynomial is stable. Recent connections between stability of polynomials and the strong Rayleigh property revealed by Brändén allows for a computationally feasible check of stability for multi-affine polynomials. Using this method we obtain a simpler proof for the n=3 case of the MCP conjecture, and a new proof for the n=4 case. We also show a multivariate version of the stability of Eulerian polynomials for n≤5 which arises as a special case of the multivariate MCP conjecture.