  From the point of view of canonical quantum gravity, it has become imperative
to find a framework for quantization which provides a {\em general}
prescription to find the physical inner product, and is flexible enough to
accommodate non-canonical variables. In this dissertation I consider an
algebraic formulation of the Dirac approach to the quantization of constrained
systems, due to A. Ashtekar. The Dirac quantization program is augmented by a
general principle to find the inner product on physical states. Essentially,
the Hermiticity conditions on physical operators determine this inner product.
I also clarify the role in quantum theory of possible algebraic identities
between the elementary variables. I use this approach to quantize various
finite dimensional systems. Some of these models test the new aspects of the
algebraic framework. Others bear qualitative similarities to \gr, and may give
some insight into the pitfalls lurking in \qg. In (spatially compact) general
relativity, the Hamiltonian is constrained to vanish. I present various
approaches one can take to obtain an interpretation of the quantum theory of
such ``dynamically constrained'' systems. I apply some of these ideas to the
Bianchi I cosmology, and analyze the issue of the initial singularity in
quantum theory.
