Topics in Grand Unified Theories: i) The Naturalness Problem. ii) Monopoles and Fermion Number Violation

This dissertation consists of two parts. The first part contains a discussion of the 'fine-tuning' and 'naturalness ' problems in grand unified theories. It is argued that, while it is impossible to solve these problems in conventional theories which contain scalars, supersymmetric theories that require no fine tuning can be constructed. In these theories the problem reduces to that of obtaining a light Higgs doublet at the tree level, without any unnatural adjustment of parameters. A realistic supersymmetric grand unified theory that has this feature is constructed. It is based on the gauge group SO(10). Supersymmetry is explicitly broken through terms of dimension two. The second part is an analysis of the interaction of fermions with a non-Abelian ('t Hooft-Polyakov) monopole. Monopoles are invariably present in grand unified theories, and recent studies with massless isospin half fermions have shown that monopoles catalyse fermion number violation. We show that this phenomenon can be described in simple terms using the language of instanton physics. This description also permits a straightforward extension of previous results to arbitrary fermion representations. The importance of half-integer winding numbers is stressed. An explicit calculation is done in the case of isovector fermions.