In free probability the notion of free convolution of probability distributions on  has played an important role since its inception by D.

Voiculescu some  years ago. In 2013, Voiculescu generalized the notion of free independence to study left and right actions on reduced free product spaces simultaneously, known as bi-free independence. One generalization of the free convolution to the bi-free setting is the bi-free convolution of planar probability distributions. In this talk, we will explain that the bi-freely infinitely divisible laws, and only these laws, can be used to approximate the distributions of sums of identically distributed bi-free pairs of commuting faces. We will also talk about bi-free Lévy-Khintchine representations from an infinitesimal point of view. The proofs depend on the bi-free harmonic analysis machinery that we developed for integral transforms of two variables, and the combinatorics of moments and bi-free cumulants. If time permits, some recent developments in this direction will also be discussed.