Level set methods have gained popularity in a number of research areas from computational physics to computer vision to computer graphics. They provide for a robust representation of geometry that can be dynamically evolved by solving partial differential equations in a manner that has been fine tuned by a community of researchers over a number of years. Traditionally, simulation of natural phenomena has been the focus of the computational physics community, but more recently computer graphics researchers have become interested in these methods in part because water, smoke and fire are exciting elements in high demand for special effects in feature films. Although computer graphics is the "inverse problem" to computer vision and this chapter is genuinely more concerned with synthesis than acquisition, understanding the application of level set methods to the physics problems that motivated their invention should enable computer vision researchers to gain a deeper understanding and appreciation of these methods and the related underlying mathematics.