  Stochastic growth phenomena on curved interfaces are studied by means of
stochastic partial differential equations. These are derived as counterparts of
linear planar equations on a curved geometry after a reparametrization
invariance principle has been applied. We examine differences and similarities
with the classical planar equations. Some characteristic features are the loss
of correlation through time and a particular behaviour of the average
fluctuations. Dependence on the metric is also explored. The diffusive model
that propagates correlations ballistically in the planar situation is
particularly interesting, as this propagation becomes nonuniversal in the new
regime.
