We begin by discussing the natural diffusion associated to mean curvature flow and work of Soner and Touzi showing that, in Euclidean space, this stochastic structure allows one to reformulate mean curvature flow as the solution to a type of stochastic target problem. Then we describe work with Ionel Popescu adapting the target problem formulation to Ricci flow on compact surfaces and using the accompanying diffusion to understand the convergence of the normalized Ricci flow. We aim to avoid being overly technical, instead focusing on the ideas underlying the appearance of stochastic objects in the context of curvature flow.