I will talk about recent joint work with Jochen Bruening and Gilles Carron. We consider complete noncompact Riemannian manifolds of finite volume with finitely many ends of type $[0,\infty)\times N$, where $N$ is closed and connected, and the metric along the ends is strictly negatively pinched of the form $dt^2+g_t$, where $g_t$ decays. We investigate whether Dirac operators on such manifolds are Fredholm operators and, if yes, try to determine their index.