In this paper, a computational method for solving fluid flow problems with moving interfaces is presented. Herein, adaptive quadtree grids are used coupled with the CICSAM  free surface capturing Volume of Fluid (VoF) method and PLIC reconstruction to interpolate the volume fraction field during refinement and derefinement processes. The combination of high resolution adaptive hierarchical remeshing and CICSAM interface advection is shown to overcome the problems of interface smearing and high CPU intensivity inherent in most VoF schemes. The result is a combination of free surface tracking and free surface capturing in that the interface is effectively tracked by the adapting refinements in the quadtree grid. In this way, a sharp interface is achieved and the advantages of both free surface tracking and capturing are combined. The new method is applied to interface advection examples in translating, rotating and shearing flow fields and the benefits of using adapting quadtree grids demonstrated.