  These are notes for a series of lectures presented at the ASIDE conference
2016. The definition of a cluster algebra is motivated through several
examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the
appearance of double Bruhat cells in the theory of total positivity. Once the
definition of cluster algebras is introduced in several stages of increasing
generality, proofs of fundamental results are sketched in the rank 2 case. From
these foundations we build up the notion of Poisson structures compatible with
a cluster algebra structure and indicate how this leads to a quantization of
cluster algebras. Finally we give applications of these ideas to integrable
systems in the form of Zamolodchikov periodicity and the pentagram map.
