In this review we will study the phase space of cosmologies consisting of a scalar field with an exponential potential and a barotropic fluid. This is a very simple system which, however, gives rise to many interesting solutions that can be classified according to the values of the parameters in the game, namely those that define the type of fluid and the steepness of the potential. We shall then pay particular attention to scaling solutions, where the scalar energy density tracks that of the fluid.
In the second part of this review we shall turn to specific models in order to apply the previous results. The goal is to study the dynamics of scalar fields in the Early Universe, and the framework to do so is different types of string compactifications. The corresponding scalar fields are then the so-called moduli for which non perturbative potentials, of an exponential type, usually arise. We shall see how, contrary to previous expectations, the existence of scaling in this context allows these fields to dynamically stabilise at their tiny minima for a large fraction of initial conditions. This represents a further step in the quest for finding a suitable Particle Physics model compatible with the existing cosmology.