We will discuss some probabilistic aspects of random lozenge tilings, the statistical mechanics generalization of the usual plane partitions from algebraic combinatorics. We will show how to study their limiting behavior (as the grid size goes to 0) using symmetric functions. Results include the fact that the positions of the horizontal lozenges near a flat vertical boundary have the same distribution as the eigenvalues of matrices from the Gaussian Unitary Ensemble; and the existence of a limit shape for symmetric plane partitions. Partially based on joint work with V. Gorin.