The paper contains a review of results which can be obtained by applying the Witten deformation method and general semi-classical asymptotics to the case of regular covering manifolds. The author formulates general semi-classical asymptotics and Morse type inequalities in a way which is more explicit, making use of the general notion of a model operator. For example, the simplest Morse inequalities can be proved using the Witten deformation associated to a Morse function applied to the Laplacian on exterior \(p\)-forms on a compact Riemannian manifold. The author then studies noncompact regular covering spaces and uses a von Neumann dimension instead of the usual one to obtain more general \(L^2\)-Morse inequalities. He also derives the \(L^2\) version of the Novikov inequalities for vector fields. The paper contains a very nice historical section and a very good bibliography devoted to the subject.