Analysis on fibred cusp spaces

We give a survey of analytic and geometric results on `fibred cusp spaces', a large class of non-compact Riemannian manifolds which include the regular parts of singular spaces with incomplete cusp singularities as well as complete spaces with asymptotically hyperbolic cusp or asymptotically Euclidean structures at infinity. These results cover topics in spectral geometry, in particular analytic torsion and index theory, and boundary value problems. The underlying tools include a careful microlocal analysis of the resolvent and the heat kernel. We include an exposition of the geometric and analytic foundations and sketch the ideas of the proofs of the main theorems. Special emphasis is put on the common features of and the differences between the incomplete and various kinds of complete settings.