This thesis discusses some existence and uniqueness problems of harmonic maps. It consists of two parts: Part I. Existence of harmonic maps with prescribed finite singularities. Here we address the question of existence of a harmonic map from a spatial domain to the sphere S\sp2 which has a prescribed finite set of singularities. Part II. Uniqueness of energy minimizing harmonic maps for almost all smooth boundary data. Suppose Ω is a smooth domain in R\spm and N is a compact smooth manifold. Here we show roughly that almost all smooth maps from ∂Ω to N serve as boundary values for a unique energy minimizing map u from Ω to N. This involves constructing a finite measure on a suitable (infinite dimensional) space of smooth boundary values.