Abstract : I present a joint work with S. Popa and D. Shlyakhtenko introducing a cohomology theory for quasi-regular inclusions of von Neumann algebras. In particular, we define $L^2$-cohomology and $L^2$-Betti numbers for such inclusions. Applying this to the symmetric enveloping inclusion of a finite index subfactor, we get a cohomology theory and a definition of $L^2$-Betti numbers for finite index subfactors, as well as for arbitrary rigid $C^*$-tensor categories. For the inclusion of a Cartan subalgebra in a $II_1$ factor, we recover Gaboriau's $L^2$-Betti numbers for equivalence relations.