In recent years, the use of Peter-Weyl theory (the theory of Fourier analysis on compact Lie groups) to define so-called ``global symbols'' of operators on compact Lie groups has emerged as a fruitful technique to study pseudo-differential operators. The aim of this thesis is to discuss similar techniques in the setting of compact homogeneous spaces.

The approach is to relate operators on homogeneous spaces to those on compact Lie groups, and then to utilize the recently developed techniques on such groups.

Two methods of associating operators on homogeneous spaces with those on compact Lie groups, called projective and horizontal lifting, along with their properties, merits and problems are considered. A key tool used in this analysis is the notion of a difference operator. This thesis includes a detailed study of such operators and their properties, combined with comprehensive calculations involving such operators on the homogeneous spaces $\Sbb^{n-1} = \SO(n)/\SO(n-1)$.

This thesis concludes with a generalization of the symbolic calculus on compact Lie groups developed by M. Ruzhansky and V. Turunen together with a collection of conjectures, which if proven would relate the generalization to pseudo-differential theory on homogeneous spaces.