Permutation limit theory arises by viewing a permutation as a probability measure on the unit square, and is motivated by dense graph limit theory. Using the theory of permutation limits (permutons), we can compute limiting properties of various permutation statistics for random permutations, such as number of fixed points, number of small cycles, pattern counts, and degree distribution of permutation graphs. We can also derive LDPs for random permutations. Our results apply to many non uniform distributions on permutations, including the the celebrated Mallows model, and mu-random permutations. This is based on joint work with Bhaswar Bhattacharya, Jacopo Borga, Sayan Das and Peter Winkler.