The dynamics of a continuum interface model in a random medium are studied and the results applied to the random-field Ising model. We find that if the dimensionality d<5, the interface will move only if a force beyond a finite depinning threshold Fch4(5-d) is applied, where h is the random field strength. Thus, when the random-field Ising model is quenched to low temperatures, there is a critical value Rc1h4(5-d) for the average radius of curvature R of the domain walls. If R>Rc, the domain structure is frozen. If R<Rc, the domain structure evolves until RRc. © 1984 The American Physical Society.