I will present a joint work with Paul Bourgade (Harvard) about the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices (CUE) and matrices from the Gaussian Unitary Ensemble. In particular, we show that the smallest gaps when rescaled by N-4/3, are Poissonian and we give the limiting distribution of the kth smallest gap. We also show that the largest gap, when normalized by √log N/N, converges in L^p to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.