The Ricci flow dictates a natural way to deform a geometric space using a reaction-diffusion equation. It was introduced by Hamilton and gained its importance through the years, including several applications and the resolution of the Poincaré Conjecture. As the heat flow equation spreads the heat over a surface, the Ricci flow tries to spread the Ricci curvature over the space, making its curvature more homogeneous. However, topology and geometry come into play and obstacles arise, leading us to partial or total collapses of the initial object.

In this talk we introduce some notions of the flow and study the particularly interesting case of homogeneous manifolds, i.e., of geometric spaces with a rich group of symmetries. We present a novel normalization of the flow with natural compactness properties, and study its collapses by presenting a characterization of Gromov-Hausdorff limits of homogeneous spaces. As an application, we present a detailed picture of the Ricci flow for a special class of homogeneous: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. Moreover, we achieve a full classification of the collapses in this class.