There are several questions about the behavior of Laplace eigenfunctions that are extremely hard to tackle and hence remain unsolved. Among the features that we don’t fully understand yet are: the number of critical points, the size of the zero set, the number of components of the zero set, and the topology of such components. A natural approach is then to randomize the problem and study these features for a randomized version of the eigenfunctions. In this talk I will present several results that tackle the problems described above for random linear combinations of eigenfunctions (with Gaussian coefficients) on a compact Riemannian manifold. This talk is based on joint works with Boris Hanin and Peter Sarnak.