  In this survey we collect some of the recent results on the "nodal geometry"
of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic
behavior, for high energy levels, of the nodal length of Gaussian Laplace
eigenfunctions on the torus (arithmetic random waves) and on the sphere (random
spherical harmonics). We give some insight on both Berry's cancellation
phenomenon and the nature of nodal length second order fluctuations
(non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic
components. Finally we consider the general case of monochromatic random waves,
i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a
compact Riemannian surface with frequencies from a short interval, whose
scaling limit is Berry's Random Wave Model. For the latter we present some
recent results on the asymptotic distribution of its nodal length in the high
energy limit (equivalently, for growing domains).
