  This thesis describes a new approach to conformal field theory. This approach
combines the method of coadjoint orbits with resolutions and chiral vertex
operators to give a construction of the correlation functions of conformal
field theories in terms of geometrically defined objects. Explicit formulae are
given for representations of Virasoro and affine algebras in terms of a local
gauge choice on the line bundle associated with geometric quantization of a
given coadjoint orbit; these formulae define a new set of explicit bosonic
realizations of these algebras. The coadjoint orbit realizations take the form
of dual Verma modules, making it possible to avoid the technical difficulties
associated with the two-sided resolutions which arise from Feigin-Fuchs and
Wakimoto realizations. Formulae are given for screening and intertwining
operators on the coadjoint orbit representations. Chiral vertex operators
between Virasoro modules are constructed, and related directly to Virasoro
algebra generators in certain cases. From the point of view taken in this
thesis, vertex operators have a geometric interpretation as differential
operators taking sections of one line bundle to sections of another. A
suggestion is made that by connecting this description with recent work
deriving field theory actions from coadjoint orbits, a deeper understanding of
the geometry of conformal field theory might be achieved.
