  In this paper, we obtain parametrizations of the moduli space of principal
bundles over a compact Riemann surface using spaces of Hecke modifications in
several cases. We begin with a discussion of Hecke modifications for principal
bundles and give constructions of "universal" Hecke modifications of a fixed
bundle of fixed type. This is followed by an overview of the construction of
the "wonderful," or De Concini--Procesi, compactification of a semi-simple
algebraic group of adjoint type. The compactification plays an important role
in the deformation theory used in constructing the parametrizations. A general
outline to construct parametrizations is given and verifications for specific
structure groups are carried out.
