An old result of Dougherty, Jackson and Kechris implies that the boundary action of the free group F2 induces a hyperfinite equivalence relation. During the talk, I will discuss generalizations of this theorem to the class of hyperbolic groups. The examples discussed will include groups acting properly and cocompactly on CAT(0) cube complexes. This is joint work with Jingyin Huang and Forte Shinko.