In this thesis, we will begin by analysing the domain mapping method for elliptic partial differential equations defined over random surfaces and random bulk-surface systems. In particular, we will begin by deriving expressions for the pull-back of geometric quantities and tangential differential operators defined over a random surface, onto a deterministic reference surface via a prescribed domain mapping. These calculations will allow for the original considered elliptic equations posed either over a random surface or a random bulk-surface system, to be reformulated respectively onto a deterministic reference surface and a deterministic bulk-surface system, and lead to the consideration of stochastic elliptic equations posed over a deterministic domain. An abstract analysis will subsequently be presented to treat the arising equations, and a numerical scheme based upon a piecewise linear finite element discretisation and a linear approximation of the curved reference domain will be presented and analysed in the abstract setting. Optimal error estimates will be derived and the convergence rates will be numerical verified in the case of a model elliptic surface equation and a coupled bulk-surface system. In the following chapter, we extend the application of the domain mapping method to the consideration of advection-diffusion equations posed over randomly evolving surfaces and randomly evolving bulk-surface systems. This will similarly entail first deriving expressions for the pullback of time-dependent quantities, such as the material derivative, onto the reference domain, which will allow for a reformulation of the considered partial differential equations posed over the random domain, onto the reference domain to take place. After which, an abstract analysis of the stochastic partial differential equations which arise after reformulating the original advection-diffusion equations onto the deterministic reference domain will be presented. A numerical scheme based upon a piecewise linear finite element approximation coupled with a single level Monte-Carlo sampling, will subsequently be presented and analysed in the abstract setting and optimal error estimates will be derived. The convergence rate are subsequently numerically verified. The thesis will then conclude with a future outlook on the applications of the domain mapping method, in particular examining how the domain mapping method may applied to a Hele-Shaw problem and a two-phase Stefan problem both posed over a random surface.