  The aim of this paper is to obtain an estimation of Hausdorff as well as
fractal dimensions of random attractors for a class of stochastic partial
differential equations with delay. The stochastic equation is first transformed
into a delayed random partial differential equation by means of a random
conjugation, which is then recast into an auxiliary Hilbert space. For the
obtained equation, it is firstly proved that it generates a random dynamical
system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation
possesses random attractors by a uniform estimate of the solution and the
asymptotic compactness of the generated RDS. After establishing the variational
equation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surely
differentiable properties of the RDS, an upper estimate of both Hausdorff and
fractal dimensions of the random attractors are obtained.
