How does a uniformly random tiling of a very large domain look? This question has fascinated mathematicians for decades, partly since it highlights a central phenomenon in physics: local behaviors of highly correlated systems can be very sensitive to boundary conditions. Indeed, a salient feature of random tiling models is that the local densities of tiles can differ considerably in different regions of the domain, depending on the boundary data. Thus, a question of interest, originally mentioned by Kasteleyn in 1961, is how the shape of the domain affects the local behavior of a random tiling. In this talk, we outline recent work that provides an answer (originally predicted by Cohn-Kenyon-Propp in 2001) to this question for random lozenge tilings of essentially arbitrary domains.