  We provide a detailed analysis of Friedmann-Robertson-Walker universes in a
wide range of scalar-tensor theories of gravity. We apply solution-generating
methods to three parametrised classes of scalar-tensor theory which lead
naturally to general relativity in the weak-field limit. We restrict the
parameters which specify these theories by the requirements imposed by the
weak-field tests of gravitation theories in the solar system and by the
requirement that viable cosmological solutions be obtained. We construct a
range of exact solutions for open, closed, and flat isotropic universes
containing matter with equation of state $p\leq \frac{1}{3}\rho$ and in vacuum.
We study the range of early and late-time behaviours displayed, examine when
there is a `bounce' at early times, and expansion maxima in closed models.
