Fluid pinch-off is possibly the simplest example of a finite-time singularity in a physical system. Quantities such as pressure and velocity diverge to infinity while length scales shrink to zero. The most ubiquitous example of pinch-off is a leaking faucet or the dripping of rain from the tip of a leaf. This work comprises the author`s PhD thesis, which classifies pinch-off in a number of systems including bubbles, droplets, and floating liquid lenses. Novel experiments using high-speed video and electronics are fully detailed, the results of which have been published in numerous professional journals. In addition, numerical approaches to simulating fluid pinch-off are discussed and implemented using Mathematica. Although the intended reader is a senior undergraduate or graduate student in physics, scientists and engineers alike will find this a useful text for experimental design and analysis in the field of fluid mechanics.