The interface-tracking and interface-capturing techniques we developed in recent years for computation of flow problems with moving boundaries and interfaces rely on stabilized formulations such as the streamline-upwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) methods. 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, typically used with non-moving meshes, are based on a stabilized semi-discrete formulation of the Navier–Stokes equations, combined with a stabilized formulation of the advection equation governing the time-evolution of an interface function marking the interface location. We provide an overview of the interface-tracking and interface-capturing techniques, and highlight how we determine the stabilization parameters used in the stabilized formulations. Copyright © 2003 John Wiley & Sons, Ltd.