Ancient solutions of the mean curvature flow and Ricci flow arise naturally as blowup limits at singularities. In recent years there has been a lot of progress towards the classification of such solutions. We will discuss some uniqueness results for convex ancient solutions of mean curvature flow, and ancient solutions of Ricci flow which have nonnegative curvature operator. In both cases we show that, under an optimal curvature growth assumption (Type I growth), the solution evolves self-similarly by scaling. The arguments for the two flows are surprisingly similar, despite being completely extrinsic and intrinsic, respectively.