We present an overview of experiments, numerical simulations, and mathematical analysis of the breakup of a low viscosity drop inside a viscous fluid. The breakup of a drop or bubble in a fluid is one of the simplest examples of a singularity in a physical system. We show the breakup of a bubble with negligible interior viscosity exhibits an exceptional form of singularity, in which the final shape at breakup retains an imprint of the initial and boundary conditions. We also consider the role of surface contaminants, or surfactants, on the dynamics near breakup, focusing on two examples. In the first example, a bubble evolves solely under the action of surface tension in a quiescent fluid. The usual mode of clean bubble breakup is changed by surfactant and a slender, quasisteady thread forms connecting two parent bubbles. The dynamics is elucidated by a combination of direct numerical simulation and long wave asymptotic analysis. In the second example, a surfactant-laden bubble is stretched by a steady extensional flow. Asymptotic analysis based on the slenderness of the bubble shows the formation of experimentally observed tip-streaming filaments at sufficiently large strain rates. In both examples, the role of effects such as small interior fluid viscosity and small diffusion of both adsorbed and dissolved surfactant is discussed.