For a discrete quantum group G, I will introduce the concept of a G-boundary action on a C*-algebra. It turns out that there exists a largest (with respect to G-covariant completely isometric embeddings) G-boundary, which we call the noncommutative Furstenberg boundary of G. For a discrete quantum group G of Kac type, I will show that the unique trace property of C*(G) follows from the the faithfulness of the G-action on the noncommutative Furstenberg boundary of G. Then, I will discuss the Gromov boundary of a generic free orthogonal quantum group, showing that this is indeed a boundary in our sense, and that the corresponding action is faithful. Joint work in progress with Kalantar, Skalski and Vergnioux.