What do permutations of 1 through n, for large n, look like?  For example,  how can we generate a random permutation that inverts a third of its pairs?   How many such permutations are there?     Permutons are doubly-stochastic measures; they are exactly the limit objects for large permutations, in the appropriate topology.  By finding permutons that maximize entropy, we (with Rick Kenyon, Dan Kral and  Charles Radin) are able in some cases to count and describe permutations  with specified pattern densities.  We'll show how permutons arise in several situations, and try to explain why they work, in some respects, better than graphons do for graphs.