  One of the remarkable properties of cluster algebras is that any cluster,
obtained from a sequence of mutations from an initial cluster, can be written
as a Laurent polynomial in the initial cluster (known as the "Laurent
phenomenon"). There are many nonlinear recurrences which exhibit the Laurent
phenomenon and thus unexpectedly generate integer sequences. The mutation of a
typical quiver will not generate a recurrence, but rather an erratic sequence
of exchange relations. How do we "design" a quiver which gives rise to a given
recurrence? A key role is played by the concept of "periodic cluster mutation",
introduced in 2009.
  Each recurrence corresponds to a finite dimensional map. In the context of
cluster mutations, these are called "cluster maps". What properties do cluster
maps have? Are they integrable in some standard sense?
  In this review I describe how integrable maps arise in the context of cluster
mutations. I first explain the concept of "periodic cluster mutation", giving
some classification results. I then give a review of what is meant by an
integrable map and apply this to cluster maps. Two classes of integrable maps
are related to interesting monodromy problems, which generate interesting
Poisson algebras of functions, used to prove complete integrability and a
linearisation. A connections to the Hirota-Miwa equation is explained.
