There are several questions about the behavior of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of their number of critical points, the study of the size of their zero set, the study of the number of connected components of their zero set, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for random linear combinations of eigenfunctions. In this talk I will present several results in this direction. This talk is based on joint works with Boris Hanin and Peter Sarnak.