In this work we focus on numerical methods for problems emerging from interfacial physics, where complex and potentially moving interfaces play a central role. In particular we present a variety of novel computational algorithms that can be applied to a wide variety of applications ranging from the solidification of binary alloys to turbulent fluid flow around deformable solids. One of the key feature of these algorithms is that interfaces and all the discontinuities they induce are captured in a sharp manner. Rather than presenting these algorithms in a general context, each chapter is dedicated to one specific application and the numerical approach we have developed to model it. Numerous examples are presented to illustrates the capabilities of our framework and prove ts accuracy. Even though all the methods presented here were developed for specific applications we believe that they can be applied to a much wider range of applications.