The concept of boundary actions goes back to the work of Furstenberg in the 1950s and was recently applied to the study of discrete group C*-algebras by Kalantar and Kennedy. I will present recent joint work with Kasprzak, Kalantar and Skalski where we extend this notion, and some of its operator algebraic applications, to the case of quantum groups. I will also discuss the important example of the Gromov boundary of orthogonal free quantum groups.
