Cohomological interpretation of automorphic functions

Bruggeman, Lewis and Zagier provided a cohomological interpretation of Maass wave forms for hyperbolic surfaces of finite area, in which each construction and isomorphism is explicit. Recently Bruggeman and I extended these results to a class of hyperbolic surfaces of infinite area and to automorphic functions that are more general than Maass wave forms.  After presenting our motivation to seek for such cohomological interpretations of Laplace eigenfunctions, I will give a gentle survey of the constructions of these cohomological interpretations with an emphasis on insights, heuristics and their relation to dynamics.