  The use of continuum phase-field models to describe the motion of
well-defined interfaces is discussed for a class of phenomena, that includes
order/disorder transitions, spinodal decomposition and Ostwald ripening,
dendritic growth, and the solidification of eutectic alloys. The projection
operator method is used to extract the ``sharp interface limit'' from phase
field models which have interfaces that are diffuse on a length scale $\xi$. In
particular,phase-field equations are mapped onto sharp interface equations in
the limits $\xi \kappa \ll 1$ and $\xi v/D \ll 1$, where $\kappa$ and $v$ are
respectively the interface curvature and velocity and $D$ is the diffusion
constant in the bulk. The calculations provide one general set of sharp
interface equations that incorporate the Gibbs-Thomson condition, the
Allen-Cahn equation and the Kardar-Parisi-Zhang equation.
