We discuss the asymptotic behavior of a class of non-autonomous stochastic lattice systems driven by multiplicative white noise. We first prove the existence and uniqueness of tempered random attractors in a weighted space containing all bounded sequences, and then establish the upper semi-continuity of these attractors as the intensity of noise approaches zero. We also prove the existence of maximal and minimal tempered random complete solutions which bound the attractors from above and below, respectively. When deterministic external terms are periodic in time, we show the random attractors are pathwise periodic. Finally, we discuss a stochastic system which possesses an infinite-dimensional tempered random attractor. This is joint work with Peter W. Bates and Kening Lu.