  We consider general approach to exactly solvable 2D dilaton cosmology with
one-loop backreaction from conformal fields taken into account. It includes as
particular cases previous models discussed in literature. We list different
types of solutions and investigate their properties for simple models, typical
for string theory. We find a rather rich class of everywhere regular solutions
which exist practically in every type of analyzed solutions. They exhibit
different kinds of asymptotic behavior in past and future, including inflation,
superinflation, deflation, power expansion or contraction. In particular, for
some models the dS spacetime with a time-dependent dilaton field is the exact
solution of field equations. For some kinds of solutions the weak energy
condition is violated independent of a specific model. We find also the
solutions with a singularity which is situated in an infinite past (or future),
so at any finite moment of a comoving time the universe is singularity-free. It
is pointed out that for some models the spacetime may be everywhere regular
even in spite of infinitely large quantum backreaction in an infinite past.
