It is shown that, by imposing reparametrization invariance, one may derive a variety of stochastic equations describing the dynamics of surface growth and identify the physical processes responsible for the various terms. This approach provides a particularly transparent way to obtain continuum growth equations for interfaces. It is straightforward to derive equations which describe the coarse grained evolution of discrete lattice models and analyze their small gradient expansion. In this way, the authors identify the basic mechanisms which lead to the most commonly used growth equations. The advantages of this formulation of growth processes is that it allows one to go beyond the frequently used no-overhang approximation. The reparametrization invariant form also displays explicitly the conservation laws for the specific process and all the symmetries with respect to space-time transformations which are usually lost in the small gradient expansion. Finally, it is observed, that the knowledge of the full equation of motion, beyond the lowest order gradient expansion, might be relevant in problems where the usual perturbative renormalization methods fail.
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