Von Neumann algebras are certain *-subalgebras of bounded operators acting on a Hilbert space. They are generally thought of as non-commutative measure spaces and offer connections to many fields of mathematics (e.g. group theory, low-dimensional topology, logic, ergodic theory, and random matrix theory to name a few). In some instances an analogy with probability spaces is more appropriate, and indeed this is precisely what informs the field of free probability, wherein one uses non-commutative analogs of probabilistic notions to study the structure of von Neumann algebras. One particular example of this is free transport. In probability theory, transport refers to a measurable map between probability spaces that pushes one measure onto the other. Following work of Brenier in 1991, transportation theory has known great success. Free transport, the non-commutative analog that was introduced by Guionnet and Shlyakhtenko in 2014, offers methods for proving isomorphisms between von Neumann algebras. In this talk, I will discuss these ideas as well my work, which used free transport to prove isomorphisms between certain so-called "non-tracial" von Neumann algebras.