The numerical simulation of physical phenomena is very well understood given that the input data are given exactly. However, in practice, the collection of these data is usually subjected to measurement errors. The goal of uncertainty quantification is to assess those errors and their possible impact on simulation results. In this talk, we address different numerical aspects of uncertainty quantification in elliptic partial differential equations on random domains. Starting from the modeling of random domains via random vector fields, we discuss how the corresponding Karhunen-Loeve expansion can efficiently be computed. Moreover, we provide a means to rigorously control the approximation error. Considering Electrical Impedance Tomography as an example, we show how measurement data can be incorporated into the model by means of Bayesian inversion. We provide Numerical results to illustrate the presented approach.