Fluids can form droplets that pinch off from each other in many situations ranging from ink jet printers to the kitchen sink. The phenomena that occur a few microseconds from pinch-off involve a balance of surface tension forces, inertia, and viscosity. Droplet pinch-off is an example of a nonlinear phenomenon that leads to finite time power law singularities and striking geometric structures that obey scaling relations and have universal behavior, in rough analogy with thermodynamics near the critical point. I will discuss the mathematics used to describe these singularities, as well as recent experiments using high speed video to study pinch-off in liquids and bubbles in 2D and 3D.