Understanding the asymptotic dynamics of the Universe is one of the main goals of theoretical cosmology. In the context of homogeneous cosmological models, where the Einstein equations of general relativity reduce to a system of ordinary differential equations, our knowledge on this problem has increased substantially over the years, thanks to the application of methods from the theory of (finite dimensional) dynamical systems. The purpose of this talk is twofold: Firstly, some classical result on the dynamics of homogeneous, Bianchi I perfect fluid cosmological models will be reviewed. Secondly the generalization of these results when the perfect fluid matter model is replaced by an elastic material will be presented. In the latter case a completely new behaviour for Bianchi type I solutions is found in the limit toward the initial (big-bang) singularity, namely the existence of oscillations between different type of vacuum solutions. Some important open problems, offering challenging questions to the dynamical systems comunity, will be addressed.