  We study pullback attractors of non-autonomous non-compact dynamical systems
generated by differential equations with non-autonomous deterministic as well
as stochastic forcing terms. We first introduce the concepts of pullback
attractors and asymptotic compactness for such systems. We then prove a
sufficient and necessary condition for existence of pullback attractors. We
also introduce the concept of complete orbits for this sort of systems and use
these special solutions to characterize the structures of pullback attractors.
For random systems containing periodic deterministic forcing terms, we show the
pullback attractors are also periodic. As an application of the abstract
theory, we prove the existence of a unique pullback attractor for
Reaction-Diffusion equations on $\R^n$ with both deterministic and random
external terms. Since Sobolev embeddings are not compact on unbounded domains,
the uniform estimates on the tails of solutions are employed to establish the
asymptotic compactness of solutions.
