In this second edition, a comprehensive review is given for path integration in two- & three-dimensional (homogeneous) spaces of constant & non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian & in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition.

The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles & spheres) in two- & three-dimensional flat & hyperbolic spaces. In particular, the discussion of integrable billiards in circles & spheres (flat & hyperbolic space) & in three dimensions are new in comparison to the first edition.

In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics & string theory, & some further results derived from the Selberg (super-) trace formula.