I will start by giving a gentle introduction to the cluster algebras of Fomin and Zelevinsky.
Then I will explain recent results giving combinatorial formulas for cluster variables in any
cluster algebra arising from a triangulated surface, as well as some cluster algebras obtained by "folding."
This proves the positivity conjecture of Fomin and Zelevinsky for all such cluster algebras,
including all (non-exceptional) finite and affine types. This is joint work with Gregg Musiker and Ralf Schiffler.