Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface. In addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow which is a natural geometric heat flow of submanifolds. In the talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.