In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for surfaces in Euclidean space.
In this talk, which will assume no prior knowledge, I will present recent progress in classifying ancient solutions to the mean curvature flow (including joint work with Kyeongsu Choi, Robert Haslhofer and Brian White). I will also explain how this classification assists in answering fundamental questions regarding the singularity formation of the flow, and describe what are the remaining challenges in converting the mean curvature flow into the powerful tool we hope it can become.