  We study a class of linear first and second order partial differential
equations driven by weak geometric $p$-rough paths, and prove the existence of
a unique solution for these equations. This solution depends continuously on
the driving rough path. This allows a robust approach to stochastic partial
differential equations. In particular, we may replace Brownian motion by more
general Gaussian and Markovian noise. Support theorems and large deviation
statements all became easy corollaries of the corresponding statements of the
driving process. In the case of first order equations with Gaussian noise, we
discuss the existence of a density with respect to the Lebesgue measure for the
solution.
