  We discuss in detail the uniform discretization approach to the quantization
of totally constrained theories. This approach allows to construct the
continuum theory of interest as a well defined, controlled, limit of well
behaved discrete theories. We work out several finite dimensional examples that
exhibit behaviors expected to be of importance in the quantization of gravity.
We also work out the case of BF theory. At the time of quantization, one can
take two points of view. The technique can be used to define, upon taking the
continuum limit, the space of physical states of the continuum constrained
theory of interest. In particular we show in models that it agrees with the
group averaging procedure when the latter exists. The technique can also be
used to compute, at the discrete level, conditional probabilities and the
introduction of a relational time. Upon taking the continuum limit one can show
that one reproduces results obtained by the use of evolving constants, and
therefore recover all physical predictions of the continuum theory. This second
point of view can also be used as a paradigm to deal with cases where the
continuum limit does not exist. There one would have discrete theories that at
least at certain scales reproduce the semiclassical properties of the theory of
interest. In this way the approach can be viewed as a generalization of the
Dirac quantization procedure that can handle situations where the latter fails.

General Relativity and Quantum Cosmology