  We study 4d Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies obtained from
time-dependent compactifications of Type IIA 10d supergravity on various
classes of 6d manifolds (Calabi-Yau, Einstein, Einstein-K\"{a}hler). The
cosmologies we present are universal in that they do not depend on the detailed
features of the compactification manifold, but only on the properties which are
common to all the manifolds belonging to that class. Once the equations of
motion are rewritten as an appropriate dynamical system, the existence of
solutions featuring a phase of accelerated expansion is made manifest. The
fixed points of this dynamical system, as well as the trajectories on the
boundary of the phase space, correspond to analytic solutions which we
determine explicitly. Furthermore, some of the resulting cosmologies exhibit
eternal or semi-eternal acceleration, whereas others allow for a parametric
control on the number of e-foldings. At future infinity, one can achieve both
large volume and weak string coupling. Moreover, we find several smooth
accelerating cosmologies without Big Bang singularities: the universe is
contracting in the cosmological past ($T<0$), expanding in the future ($T>0$),
while in the vicinity of $T=0$ it becomes de Sitter in hyperbolic slicing. We
also obtain several cosmologies featuring an infinite number of cycles of
alternating periods of accelerated and decelerated expansions.
