Cluster algebras are commutative algebras having distinguished "clusters" of generators, with different clusters being related to each other by "mutations". Despite the terminology they are relatively elementary to define (and do not require extensive background), and come equipped with a wealth of nice properties.
Since their definition by Fomin and Zelevinsky (2002), cluster algebras have found numerous applications in geometry, representation theory, physics, and elsewhere. We will provide a brief tour of this theory and some of its many applications.