  In this work, we review the results of Refs [1]-[5] dedicated to the
description of the early Universe cosmology induced by quantum and thermal
effects in superstring theories. The present evolution of the Universe is
described very accurately by the standard Lambda-CDM scenario, while very
little is known about the early cosmological eras. String theory provides a
consistent microscopic theory to account for such missing epochs. In our
framework, the Universe is a torus filled with a gas of superstrings. We first
show how to describe the thermodynamical properties of this system, namely
energy density and pressure, by introducing temperature and supersymmetry
breaking effects at a fundamental level by appropriate boundary conditions.
  We focus on the intermediate period of the history: After the very early
"Hagedorn era" and before the late electroweak phase transition. We determine
the back-reaction of the gas of strings on the initially static space-time,
which then yields the induced cosmology. The consistency of our approach is
guaranteed by checking the quasi-staticness of the evolution. It turns out that
for arbitrary initial boundary conditions at the exit of the Hagedorn era, the
quasi-static evolutions are universally attracted to radiation-dominated
solutions. It is shown that at these attractor points, the temperature, the
inverse scale factor of the Universe and the supersymmetry breaking scale
evolve proportionally. There are two important effects which result from the
underlying string description. First, initially small internal dimensions can
be spontaneously decompactified during the attraction to a radiation dominated
Universe. Second, the radii of internal dimensions can be stabilized.
