  In this thesis quadratic and cubic algebras, which are extensions of SU(1,1)
and SU(2) are studied in detail, with particular attention being given to their
construction, their finite and infinite dimensional irreducible representations
and the differential realizations of their generators in the space of analytic
functions. The above construction is extended to formulate a general method for
the construction and representations of higher order polynomial algebras by a
generalized Jordan-Schwinger method. Various coherent states corresponding to
polynomial algebras are constructed. Physical applications of the polynomial
algebra are given.
