In the scope of this project we consider Sobolev spaces with values in Banach spaces. Those spaces are nontrivial generalizations of classical Sobolev spaces and occur naturally while treating partial differential equations. First of all we are going to describe the mappings which induce isomorphisms of Banach valued Soboev spaces. Then we develop the theory of composition operators on this space. Next mixed norm spaces will be considered and obtained some applications to the theory of partial differential equations.