Ergodic theory is a branch of pure mathematics that investigates the "chaotic" properties of dynamical systems. The time evolution of even very simple systems can be completely unpredictable, and one of the key objectives of ergodic theory is to identify and classify measures that are invariant under the time evolution, thus allowing deep insights in the structure of chaotic dynamics. Simple examples of chaotic dynamical systems include geodesic flows on negatively curved surfaces, and billiard tables with convex scatterers (Sinai billiards). Ergodic theory has provided powerful tools to solve some outstanding problems in other research fields, e.g., in number theory, combinatorics, quantum chaos and statistical physics.