T he book was written thinking to showed to the researchers in nonlinear generalized functions (or Colombeau"s full algebras) algebraic, topological, and analytical aspects of this area so important when we study nonlinear differential equations ordi nary or/and partial. In order to make this, we develop the ring of Colombeau"s full generalized numbers, we endow these numbers of topology, the sharp scalar topology, and show some results of convergences. Next, we develop the sharp topology on the ring of the Colombeau"s full generalized functions, highlight that, the compact support is dense in this space of functions, and that this metric space is complete. With these topological results, we answer a boundary-initial value problem of the par abolic type what shown the efficacy of the theory developed.