Free probability is a non-commutative probability theory that studies the joint moments of operators acting on reduced free products of vector spaces. Since its inception by Voiculescu in the 1980s as an attempt to solve the isomorphic free group factor problem, free probability has become an important part of the theory of operator algebras with several applications to random matrix theory.

In this talk, we will provide a brief overview of the recent extension of free probability known as bi-free probability. Bi-free independence generalizes the notion of free independence in order to simultaneously study the left and right regular representations on free products of vector spaces. Although adding this additional algebra provides greater flexibility in the objects that maybe studied, so far much of the theory of free probability extends to the bi-free setting. This overview will touch on many of the basic and current developments in the theory.