Take a residue, leave a residue

Residue calculus is, of course, a very powerful way to evaluate many different kinds of integrals. For multivariate integrals, however, it may get really cumbersome, to the point that one sometimes gets stuck in purely combinatorial complexity. I will discuss some ideas that have recently been used with some success to circumvent such problems. Of course, they are much more specialized than residue calculus and apply in only very special situations. Still, these situations are not without interest in concrete applications, including solutions of q-difference equations and some questions in analysis of automorphic forms.