In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.