  We investigate how a clean continuous phase transition is affected by
spatio-temporal disorder, i.e., by an external perturbation that fluctuates in
both space and time. We derive a generalization of the Harris criterion for the
stability of the clean critical behavior in terms of the space-time correlation
function of the external perturbation. As an application, we consider diffusive
disorder, i.e, an external perturbation governed by diffusive dynamics, and its
effects on a variety of equilibrium and nonequilibrium critical points. We also
discuss the relation between diffusive disorder and diffusive dynamical degrees
of freedom on the example of model C of the Hohenberg-Halperin classification,
and we comment on Griffiths singularities.
