Given a planar triangular lattice domain R, one can ask how many lozenge tilings of R exist, or if one even exists at all. These questions can be answered by associating a tiling of R both with a height function defined on the lattice and with a perfect matching of its dual graph. Upon introducing a probability measure on these tilings, one finds that random height functions converge to a deterministic limit shape as the tile size decreases. In this talk, we will discuss how to compute this limit shape as the solution of a variational problem. We will also introduce the Kenyon-Okounkov conjecture on convergence of fluctuations of centered height functions to the Gaussian free field, a random generalized function on a region equipped with a certain complex structure.