A very important (hard but tractable) open problem concerns the meaningful classification of convex ancient solutions to mean curvature flow and related flows.

Ancient solutions to geometric flows (such as the mean curvature, Ricci, or Yamabe flows) are solutions which have existed for an infinite amount of time in the past, and as such are expected to be quite rigid (diffusion has had an infinite amount of time to "sort things out"). 

Ancient solutions also model the ultra-violet regime in certain quantum field theories, and early research on ancient solutions was undertaken by physicists in this context.

They also arise naturally (through "blow-up" procedures) in the study of singularities of the flow, and a deep understanding of them will have profound implications for the continuation of the flow through singularities (a prerequisite for many important applications). In the context of extrinsic geometric flows such as the mean curvature flow, the study of convex ancient solutions (i.e. solutions whose timeslices bound convex regions in space) is particularly pertinent, since blow-ups are guaranteed to be convex in many settings. This is fortunate, since it is this class of ancient solutions which appears to be most "rigid"!

Many interesting questions regarding ancient solutions to geometric flows, ranging widely in difficulty, remain open.