The most widely used tool for the solution of optimal control problems is the Pontryagin Maximum Principle. But the Maximum Principle is, in general, only a necessary condition for optimality. It is therefore desirable to have supplementary conditions, for example second order sufficient conditions, which confirm optimality (at least locally) of an extremal arc, meaning one that satisfies the Maximum Principle. Standard second order sufficient conditions for optimality, when they apply, yield the information not only that the extremal is locally minimizing, but that it is also locally unique. There are problems of interest, however, where minimizers are not locally unique, owing to the fact that the cost is invariant under small perturbations of the extremal of a particular structure (translations, rotations or time-shifting). For such problems the standard second order conditions can never apply. The first contribution of this thesis is to develop new second order conditions for optimality of extremals which are applicable in some cases of interest when minimizers are not locally unique. The new conditions can, for example, be applied to problems with periodic boundary conditions when the cost is invariant under time translations. The second order conditions investigated here apply to normal extremals. These extremals satisfy the conditions of the Maximum Principle in normal form (with the cost multiplier taken to be 1). It is, therefore, of interest to know when the Maximum Principle applies in normal form. This issue is also addressed in this thesis, for optimal control problems that can be expressed as calculus of variations problems. Normality of the Maximum Principle follows from the fact that, under the regularity conditions developed, the highest time derivative of an extremal arc is essentially bounded. The thesis concludes with a brief account of possible future research directions.