Many scientific and engineering problems involve interconnected moving interfaces separating different regions, including dry foams, crystal grain growth and multi-cellular structures in man-made and biological materials. Producing consistent and well-posed mathematical models that capture the motion of these interfaces, especially at degeneracies, such as triple points and triple lines where multiple interfaces meet, is challenging. In this talk, we will introduce a new computational method for tracking the interface in general multiphase problems. It combines properties of level set methods with a geometric construction, yielding a robust, accurate and efficient numerical method that automatically deals with evolution of triple points/lines and topological change in the multiphase system. We present applications in geometric and fluid flow problems that show many of the method's virtues, including a model for liquid drainage in an unstable foam leading to thin-film interference patterns.