A phase-field approximation of an interface is a continuous function that takes two different constant values except on a narrow transition region around the interface.  Such an approximation can handle the interfacial topological changes and fluctuations.  We develop phase-field models for interface motion that is driven by both the local interfacial geometry such as mean curvature and an effective force on the interface resulting from a nonlocal field.  Examples of such fields include the electrostatic potential with different dielectric mediums separated by the interface and concentrations of actin and myosin molecules in a moving biological cell.  We report our computational results to demonstrate the success of such models. We also prove, using the Gamma convergence and matched asymptotic analysis, that the diffused interface as the phase-field approximation in fact converges to the corresponding sharp interface.