The Boltzmann equation models the evolution of a rarefied gas, in which particles undergo predominantly binary interactions, via its particle distribution function. The main focus of this talk will be a generalization of the Boltzmann equation called the binary-ternary Boltzmann equation, which in addition to binary particle interactions captures a certain type of ternary interactions. We will discuss the well-posedness of this equation, and the two main tools for obtaining such result - angular averaging estimates and moments estimates. We will show that moment estimates, in particular, are determined by the better part of the collision operator, and so the presence of both the binary and the ternary operator in the equation can yield better moment estimates than the purely binary or the purely ternary equation. The talk is based on a joint work with I. Ampatzoglou, I. M. Gamba and N. Pavlovic.