We provide an overview of the interface-tracking and interface-capturing techniques we have developed in recent years for computation of flow problems with moving boundaries and interfaces, including two-fluid interfaces. The interface-tracking techniques are based on the Deforming-Spatial-Domain/Stabilized Space-Time formulation, where the mesh moves to track the interface. The interface-capturing techniques, which were developed for two-fluid flows, are based on the stabilized formulation, over non-moving meshes, of both the flow equations and the advection equation governing the time-evolution of an interface function marking the location of the interface. For interface-capturing techniques, to increase the accuracy in representing the interface, the Enhanced-Discretization Interface-Capturing Technique (EDICT) can be used to to accomplish that goal. We also provide and overview of some of the additional ideas developed to increase the scope and accuracy of these two classes of techniques.

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