Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in three mini-courses which were held in Wykno (Pl) (1999), Recife (Br) (2000) and Ferrara (It) (2000), respectively.