Free probability is a noncommutative probability theory that describes the large-$n$ bulk behavior of several interacting random matrices, including in particular matrices whose joint distribution is a Gibbs measure for a noncommutative potential $V$. The objects modeling the large-$n$ limit are operators in a von Neumann algebra, a non-commutative analog of measure spaces. However, unlike the classical setting, the associated von Neumann algebras might not be isomorphic, and there are many analytical properties of von Neumann algebras that could pose obstructions to "transport of measure" or "change of coordinates". However, Guionnet and Shlyakhtenko showed that in some cases, optimal transport constructions can be performed in the noncommutative setting that lead to isomorphisms of the von Neumann algebras. Subsequent work has developed the foundations of free optimal transport theory in greater depth, showed its relationship with quantum optimal transport, and studied the information geometry of free entropy on the free Wasserstein space.