Understanding the phases of string theory in the strong curvature and high temperature regime, which is inaccessible to the field theory approximation, may provide insights about the physics of the very early Universe. Cosmological solutions can be described at the perturbative string level, arising as quantum or thermal instabilities of an initially flat background. Two major obstacles that typically prevent a perturbative treatment of the backreaction are the Hagedorn/tachyonic divergences that occur in such strong curvature and/or high temperature regions and the initial gravitational singularity (Big Bang), that always appears in the field theory approach. In this talk, I will present recent progress in tackling these problems within the framework of perturbative string theory. In particular, I will consider a special toy model whose high degree of symmetry may help uncover the stringy mechanism that protects the cosmological evolution from Hagedorn or gravitational type singularities.