Mean curvature flow is the most natural geometric heat flow for surfaces, with several existing applications for geometry, topology and physics. The biggest hurdle for finding even more such applications to the flow is understanding how singularities develop in the flow, andwhat is their nature. In recent years, it became clear that in order to obtain ‘’complete information’’ one should understand *all* potential blow-up limits at the singularity. Such general blow up-limits give rise to “ancient" mean curvature flows - flows that exist from time -\infty.
In this talk, I will describe an ongoing project, joint with Kyeongsu Choi, Robert Haslhofer, and partially also with Brian White, where we classify such ancient flows in a way that allows to derive information about the singularity formation of the flow.