This book concentrates mainly on the theorem of existence of periodic orbits for higher dimensional analogs of twist maps. The setting is that of a discrete variational calculus and the techniques involve Conley-Zehnder-Morse theory. They give rise to the concept of ghost tori which are of interest in the dimension 2 case (ghost circles). The debate is oriented somewhat toward the open problem of finding orbits of all (in particular, irrational) rotation vectors.