The subject of this talk is at the crossroads of functional analysis, ergodic theory and group theory. Using a construction by Murray and von Neumann (1943), ergodic actions of countable groups on probability spaces give rise to algebras of operators on a Hilbert space, called von Neumann algebras. In a joint work with Sorin Popa, we proved that such crossed product von Neumann algebras by free groups or, more generally, by hyperbolic groups have a unique Cartan subalgebra. I will explain this result and its consequences for the classification of crossed products by free groups.