Quantum ergodicity

The question of quantum ergodicity deals with the localization or delocalization properties for eigenfunctions of the laplacian, on (compact) riemannian manifolds. For manifolds with an ergodic geodesic flow, the Shnirelman theorem states that ``most'' eigenfunctions of the laplacian become fully delocalized, in the large eigenvalue limit. For manifolds of negative curvature, the Quantum Unique Ergodicity conjecture asks for a statement valid for all eigenfunctions. I will review related questions and results, and will compare with the case of spheres and flat tori. I will also describe some recent results of delocalization for eigenfunctions on large regular graphs, in particular a result which is a discrete analogue of the Shnirelman theorem.