`Old' String Theory is a theory of one-dimensional extended objects, whose vibrations correspond to excitations of various target-space field modes including gravity. It is for this reason that strings present the first, up to now, mathematically consistent framework where quantum gravity is unified with the rest of the fundamental interactions in nature. In these lectures I will give an introduction to low-energy Effective Target-Space Actions derived from conformal invariance conditions of the underlying sigma models in string theory. In this context, I shall discuss cosmology, emphasizing the role of the dilaton field in inducing inflationary scenaria and in general expanding string universes. Specifically, I shall analyse some exact solutions of string theory with a linear dilaton, and discuss their role in inducing expanding Robertson-Walker Universes. I will mention briefly pre-Big-Bang scenaria of String Cosmology, in which the dilaton plays a crucial role. In view of recent claims on experimental evidence (from diverse astrophysical sources) on the existence of cosmic acceleration in the universe today, with a positive non-zero cosmological constant (de Sitter type), I shall also discuss difficulties of incorporating such Universes with eternal acceleration in the context of critical string theory, and present scenaria for a graceful exit from such a phase.