This article reviews the role of reparametrization invariance (the inv
ariance of the properties of a system with respect to the choice of th
e co-ordinate system used to describe it) in deriving stochastic equat
ions that describe the growth of surfaces. By imposing reparametrizati
on invariance on a system, the authors identify the physical origin of
many of the terms in its growth equations. Both continuum-growth equa
tions for interfaces and equations for the coarse-grained evolution of
discrete-lattice models are derived with this method. A detailed anal
ysis of the discrete-lattice case and its small-gradient expansion pro
vides a physical basis for terms found in commonly studied growth equa
tions. The reparametrization-invariant formulation of growth processes
also has the advantage of allowing one to model shadowing effects tha
t are lost in the no-overhang approximation and to conserve underlying
symmetries of the system that are lost in a small-gradient expansion.
Finally, a knowledge of the full equation of motion, beyond the lowes
t-order gradient expansion, may be relevant in problems where the usua
l perturbative renormalization methods fail.