  We consider Friedmann-Lemaitre-Robertson-Walker cosmological models in the
framework of general scalar-tensor theories of gravity (STG) with arbitrary
coupling functions, set in the Jordan frame. First we describe the general
properties of the phase space in the case of barotropic matter fluid and scalar
field potential for any spatial curvature (flat, spherical, hyperbolic). Then
we address the question under which conditions epochs of accelerated and
super-accelerated expansion are possible in STG. For flat models filled with
dust matter (and vanishing potential) we give a necessary condition on the
coupling function of the scalar field which must be satisfied to allow
acceleration and super-acceleration. This is illustrated by a specific example.
