First introduced by Fomin and Zelevinsky, cluster algebras are commutative rings that have many combinatorial properties. They have had many applications to both mathematics and physics. In this talk, I will first introduce cluster algebras and explore some of their properties. I will then move on to their applications, starting with dilogarithm identities and then moving to integrable systems and the thermodynamic Bethe ansatz (TBA). Time permitting, I will connect some of these ideas to the ODE/IM correspondence.
