  We investigate the quantum theory of 1+1 dimensional dilaton gravity, which
is an interesting toy model of the black hole dynamics. The functional measures
of gravity part are explicitly evaluated and derive the Wheeler-DeWitt like
equations as physical state conditions. In ADM formalism the measures are very
ambiguous, but in our formalism they are explicitly defined. Then the new
features which are not seen in ADM formalism come out. A singularity appears at
$\df^2 =\kappa (>0) $, where $\kappa =(N-51/2)/12 $ and $ N$ is the number of
matter fields. At the final stage of the black hole evaporation, the Liouville
term becomes important, which just comes from the measures of the fields.
Behind the singularity the quantum mechanical region $\kappa > \df^2 >0 $
extends, where the sign of the kinetic term in the Wheeler-DeWitt like equation
changes. If $\kappa <0 $, the singularity disappears. We briefly discuss the
possibility of gravitational tunneling and the issue of the information loss.
  (Talk given at "YITP Workshop on Theories of Quantum Fields -Beyond
Perturbation-", Kyoto, Japan, 14-17 July 1992.
  Some misleading arguments in the preprint UT-Komaba 92-7 entitled "Quamtum
Theory of Dilaton Gravity in 1+1 Dimensions" are corrected. The several remarks
on the quantization are included. The difference from the other quantum theory
is clarified.
