A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the classical is ergodic, almost all of the Laplace eigenfunctions become uniformly distributed in phase space. There are similar statements which are valid for some integrable and pseudo-integrable systems, such as flat tori and rational polygons. I will give an introduction to these notions, including explanations of the undefined terms above, and describe some connections with number theory.