Dynamical systems is the study of systems that evolve with time, and ergodic theory is the branch of dynamics that studies the statistical and qualitative behavior of measurable actions on a measure space. The problems, results, and techniques of ergodic theory lie at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Recently, ergodic theory has seen a burst of activity in which ergodic theory and its techniques have been imported into combinatorics, number theory, and geometry. This authoritative volume, which contains entries from the Encyclopedia of Complexity and Systems Science, begins with an overview of the basic objects in ergodic theory, including recurrence, convergence theorems, mixing, and entropy, and continues with an overview of the recent connections with other fields of mathematics. These interactions include areas such as topological, smooth, and symbolic dynamics, but also involve topics traditionally outside the scope of ergodic theory, such as fractal geometry, number theory, and combinatorics.