  Random invariant manifolds are geometric objects useful for understanding
complex dynamics under stochastic influences. Under a nonuniform hyperbolicity
or a nonuniform exponential dichotomy condition, the existence of random
pseudo-stable and pseudo-unstable manifolds for a class of \emph{random}
partial differential equations and \emph{stochastic} partial differential
equations is shown. Unlike the invariant manifold theory for stochastic
\emph{ordinary} differential equations, random norms are not used. The result
is then applied to a nonlinear stochastic partial differential equation with
linear multiplicative noise.
