  Scalar-tensor gravitational theories are important extensions of standard
general relativity, which can explain both the initial inflationary evolution,
as well as the late accelerating expansion of the Universe. In the present
paper we investigate the cosmological solution of a scalar-tensor gravitational
theory, in which the scalar field $\phi $ couples to the geometry via an
arbitrary function $F(\phi $). The kinetic energy of the scalar field as well
as its self-interaction potential $V(\phi )$ are also included in the
gravitational action. By using a standard mathematical procedure, the Lie group
approach, and Noether symmetry techniques, we obtain several exact solutions of
the gravitational field equations describing the time evolutions of a flat
Friedman-Robertson-Walker Universe in the framework of the scalar-tensor
gravity. The obtained solutions can describe both accelerating and decelerating
phases during the cosmological expansion of the Universe.
