This dissertation is concerned with various mathematical aspects of Topological Quantum Field Theories (TQFTs) known as ChernSimons theories. Although this subject has its origins in theoretical physics, the treatment here is in terms of the axiomatic approach due to Segal and Atiyah. A key feature of the thesis is the notion of a 3tier (axiomatic) TQFT. This involves assigning a category to a closed Imanifold and a functor to a 2manifold with boundary which is viewed as a cobordism between Imanifolds. To a closed 2manifold Σ the theory assigns a vector space H_{Σ} , and to a 3manifold M the theory assigns a numerical invariant (if M is closed), a vector in H_{δM} (if M has closed boundary δM ) or a natural transformation of functors (if the boundary δM of M has a 1dimensional boundary). After a brief introduction, we introduce in chapter 1 the definition of a TQFT and that of a 3tier TQFT. We then describe the geometrical setup for ChernSimons Theory for a Lie group G and focus on the particular case of G = SU(2). Finally we describe quite concisely how it might fit into a 3tier TQFT structure. Roughly the next half of the thesis treats the specific case of ChernSimons theory for the circle group T. In chapter 2 we describe a number of interesting topological aspects of the theory. In chapter 3 we go on to show how the theory fits into 3tier TQFT framework. In the next two chapters we begin to deal with ChernSimons theories for G a noncompact group. In chapters 6 and 7 we deal with a rather more algebraic theory which is the abelian version of a theory which is meant to compute the Casson invariant for oriented homology 3spheres. For this reason we call it the abelian Cassontype theory. From the physics viewpoint, it coincides with the ChernSimons theory where G is a supergroup. This is rather difficult to motivate mathematically, so we adopt an algebraictopological definition of the theory and show it satisfies the TQFT axioms. We then go on to show how it fits into the 3tier TQFT structure. The novelty here is that the category assigned to a 1manifold is not semisimple.
