Hyperbolic groups form an important class of finitely presented groups. They are known to be of type Fn for all n, that is, they admit a classifying space with finitely many cells in all dimensions. It is natural to ask if subgroups of hyperbolic groups inherit these strong finiteness properties: For n >= 1, is there a subgroup of a hyperbolic group of type Fn, but not of type Fn+1? Brady raised this question in 1999 following his construction of examples for n=2, while the first examples for n=1 were constructed by Rips in 1982. The case n >= 3 remained open since. Here we will explain recent progress on Brady's question for n >= 3. This talk is based on joint works with Pierre Py, and with Bruno Martelli and Pierre Py.