Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant details due to either insufficient data, or measurement errors, or both. In such cases, this topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single- and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.