[en] We develop a theory of transport in Hamiltonian systems in the context of iteration of area-preserving maps. Invariant closed curves present complete barriers to transport, but in regions without such curves there are still invariant Cantor sets named cantori, which appear to form major obstacles. The flux through the gaps of the cantori is given by Mather's differences in action. This gives useful bounds on transport between regions, and a universal scaling law for one-parameter families when a curve has just broken, which agree well with numerical experiments of Chirikov and explain an apparent disagreement with results of Greene. By dividing the phase space into regions separated by the strongest barriers, and assuming the motion is mixing within them, we derive a global picture of transport, which can be used, for example, to predict confinement times and to explain longtime tails in the decay of correlations