An important problem in mean curvature flow is to find conditions on initial data that rule out 'collapsing' singularity models, such as the Grim Reaper. This is a necessary step towards establishing regularity and compactness theorems for the flow. A prevalent class of singularity models are the convex ancient solutions. We will first discuss examples and general properties of such solutions, before moving on to a universality result (proven with T. Bourni and M. Langford) which asserts that a convex ancient solution is collapsing if and only if it admits a sequence of rescalings converging to a Grim Reaper. This makes it possible to rule out collapsing via curvature pinching.