  New exact solutions of Einstein's gravity coupled to a self-interacting
conformal scalar field are derived in this work. Our approach extends a
solution-generating technique originally introduced by Bekenstein for massless
conformal scalar fields. Solutions are obtained for a
Friedmann-Robertson-Walker geometry both for the cases of zero and non-zero
curvatures, and a variety of interesting features are found. It is shown that
one class of solutions tends asymptotically to a power-law inflationary
behaviour $S(t)\sim t^p$ with $p>1$, while another class exhibits a late time
approach to the $S(t)\sim t$ behaviour of the coasting models. Bouncing models
which avoid an initial singularity are also obtained. A general discussion of
the asymptotic behaviour and of the possibility of occurrence of inflation is
provided.
