Almost all of the extensive research on mean curvature flow has been for surfaces without boundary. However, it is interesting and natural to consider MCF for surfaces with boundary. In this talk, I will describe a useful weak formulation of such flows that gives existence for all time with arbitrary initial data. Furthermore, under rather mild hypotheses on the initial surface, the moving surface remains forever smooth at the boundary, even after singularities may have formed in the interior. On the other hand, if one relaxes those hypotheses, then interesting boundary singularities can occur.

For several centuries, there were only two known examples of complete embedded minimal surfaces in R^3 with one end: the plane and the helicoid. Recently a third such surface was found: the genus-one helicoid. I will talk about we know the surface exists, and I will discuss some results and open questions that its discovery has led to.