  A Lagrangian treatment of the quantization of first class Hamiltonian systems
with constraints and Hamiltonian linear and quadratic in the momenta
respectively is performed. The ``first reduce and then quantize'' and the
``first quantize and then reduce'' (Dirac's) methods are compared. A new source
of ambiguities in this latter approach is revealed and its relevance on issues
concerning self-consistency and equivalence with the ``first reduce'' method is
emphasized. One of our main results is the relation between the propagator
obtained {\it \`a la Dirac} and the propagator in the full space, eq. (5.25).As
an application of the formalism developed, quantization on coset spaces of
compact Lie groups is presented. In this case it is shown that a natural
selection of a Dirac quantization allows for full self-consistency and
equivalence. Finally, the specific case of the propagator on a two-dimensional
sphere $S^2$ viewed as the coset space $SU(2)/U(1)$ is worked out.
