  Cluster algebras are commutative rings with a set of distinguished generators
having a remarkable combinatorial structure. They were introduced by Fomin and
Zelevinsky in 2000 in the context of Lie theory, but have since appeared in
many other contexts, from Poisson geometry to triangulations of surfaces and
Teichm\"uller theory. In this expository paper we give an introduction to
cluster algebras, and illustrate how this framework naturally arises in
Teichm\"uller theory. We then sketch how the theory of cluster algebras led to
a proof of the Zamolodchikov periodicity conjecture in mathematical physics.
