  An extension of the Dirac procedure for the quantization of constrained
systems is necessary to address certain issues that are left open in Dirac's
original proposal. These issues play an important role especially in the
context of non-linear, diffeomorphism invariant theories such as general
relativity. Recently, an extension of the required type was proposed by one of
us using algebraic quantization methods. In this paper, the key conceptual and
technical aspects of the algebraic program are illustrated through a number of
finite dimensional examples. The choice of examples and some of the analysis is
motivated by certain peculiar problems endemic to quantum gravity. However,
prior knowledge of general relativity is not assumed in the main discussion.
Indeed, the methods introduced and conclusions arrived at are applicable to any
system with first class constraints. In particular, they resolve certain
technical issues which are present also in the reduced phase space approach to
quantization of these systems.
