Free probability was introduced in the 1980’s by Voiculescu, with the aim of studying von Neumann algebras by viewing them as non-commutative probability spaces, and this analogy has proved quite powerful in operator algebra theory. In the 1990’s, Speicher was able to describe free independence using cumulants constructed from the lattice of non-crossing partitions. In this talk we will give an introduction to free probability and outline the role the non-crossing cumulants have played in describing the theory. Time permitting we will also demonstrate some more recent applications of combinatorics to free probability, such as in describing bi-free probability, type B free probability, and boolean independence.