  The large order growth of string perturbation theory in $c\le 1$ conformal
field theory coupled to world sheet gravity implies the presence of
$O(e^{-{1\over g_s}})$ non-perturbative effects, whose leading behavior can be
calculated in the matrix model approach. Recently it was proposed that the same
effects should be reproduced by studying certain localized D-branes in
Liouville Field Theory, which were constructed by A. and Al. Zamolodchikov. We
discuss this correspondence in a number of different cases: unitary minimal
models coupled to Liouville, where we compare the continuum analysis to the
matrix model results of Eynard and Zinn-Justin, and compact c=1 CFT coupled to
Liouville in the presence of a condensate of winding modes, where we derive the
matrix model prediction and compare it to Liouville theory. In both cases we
find agreement between the two approaches. The c=1 analysis also leads to
predictions about properties of D-branes localized in the vicinity of the tip
of the cigar in SL(2)/U(1) CFT with c=26.
