Mean curvature flow is the negative gradient flow of volume, so any closed hypersurface flows in the direction of steepest descent for volume and eventually becomes extinct in finite time. In most cases, the flow develops singularities before its extinction time. It is known that the asymptotic behaviors of the flow near a singularity are modeled on a special class of solutions to mean curvature flow, which are called self-shrinkers. In this talk, we will outline a program on the classification of noncompact two-dimensional self-shrinkers, and report some recent progress with an emphasis on the geometry at infinity of these self-shrinkers.