I will discuss two types of boundary actions for a family of universal discrete quantum groups. In a joint work with Vergnioux, boundaries at infinity are constructed as unital C*-algebras and are used to prove exactness of reduced C*-algebras as well as solidity (in the sense of Ozawa) of the associated von Neumann algebras. In a more recent joint work with Vander Vennet, Poisson boundaries of random walks on certain universal discrete quantum groups are identified with fields of infinite tensor products of matrix algebras.