  We present a review of the canonical quantization approach to the problem of
non-perturbative 2d dilaton gravity. In the case of chiral matter we describe a
method for solving the constraints by constructing a Kac-Moody current algebra.
For the models of interest, the relevant Kac-Moody algebras are based on
SL(2,R) X U(1) group and on an extended 2d Poincare group. As a consequence,
the constraints become free-field Virasoro generators with background charges.
We argue that the same happens in the non-chiral case. The problem of the
corresponding BRST cohomology is discussed as well as the unitarity of the
theory. One can show that the theory is unitary by chosing a physical gauge,
and hence the problem of transitions from pure into mixed sates is absent.
Implications for the physics of black holes are discussed. (Based on the talks
presented at Trieste conference on Gauge Theories, Applied Supersymmetry and
Quantum Gravity, May 1993 and at Danube '93 Workshop, Belgrade, Yugoslavia,
June 1993)
