Mass transportation problems, also known as Monge-Kantorovich problems, have been intensively studied in particular with respect to numerous and important applications and connections with other fields such as PDE, material sciences, probability and economics. One of these connections is concerned with Mather''s theory of minimal measures in Lagrangian dynamic. The main aim of this monograph is to present in a self-contained way some aspects of these connections and some related open problems as well. In particular, the main results presented are based on the notion of currents in Geometric Measure Theory. This monograph could be also considered as an introduction to the Monge-Kantorovich Theory by emphasizing the role of Geometric Measure Theory and some connections with Lagrangian dynamic.