  This thesis is concerned with the asymptotic behavior of solutions of
stochastic $p$-Laplace equations driven by non-autonomous forcing on
$\mathbb{R}^n$. Two cases are studied, with additive and multiplicative noise
respectively. Estimates on the tails of solutions are used to overcome the
non-compactness of Sobolev embeddings on unbounded domains, and prove
asymptotic compactness of solution operators in $L^2(\mathbb{R}^n)$. Using this
result we prove the existence and uniqueness of random attractors in each case.
Additionally, we show the upper semicontinuity of the attractor for the
multiplicative noise case as the intensity of the noise approaches zero.
