In this talk we shall describe the construction of Varopoulos' type extensions of L^p and BMO boundary functions in rough  domains. That is, smooth extensions of functions such that the L^p-norms of their non-tangential maximal function and the Carleson  functional of their gradients can be controlled by the norm of the boundary data. After giving the geometric motivation and a brief survey of known results, we will proceed to present a new and more general approach of constructing Varopoulos' extensions in domains with minor geometrical assumptions for the boundaries.