Doctor of Philosophy / Department of Mathematics / Louis Crane / In this thesis we explore the possibilities of obtaining Topological Quantum Field Theories using cobordisms with corners to break further down in the structure of manifolds of a given dimension. The algebraic data obtained is described in the language of higher category theory, more precisely in its cubical approach which we explore here as well. Interesting connections are proposed to some important objects in Physics: the representations of the Poincaré group. Finally we will describe in great details the topological tools needed to describe the categories of cobordisms with corners and give some conjectures on their nature.