Some of the most important problems in geometric flows are related to the understanding of singularities. In 1995 R. Hamilton introduced the notion of ancient solutions to study blow up limits at Ricci flow singularities. These are special solutions which are defined for all time −∞< t≤T, for some T≤+∞. G. Perelman, in his seminal works on the Poincare and Thurston’s geometrization conjectures, introduced the notion of κ-noncollapsed ancient solutions and showed that they model singularities of the Ricci flow in dimension three. In this talk, we will give an overview of recent works on the classification of ancient solutions to Mean curvature flow and Ricci flow. In particular, we will discuss the resolution of a conjecture by G. Perelman in the compact and non-compact cases.