This is a corrected and annotated version of the author’s seminal PhD thesis, in which two forms of asymptotically universal structure were presented and explained for area-preserving maps. Area-preserving maps are the discrete-time analogue of two degree-of-freedom Hamiltonian systems, and much of their dynamics is reviewed in Chapter 1. The asymptotically universal structure is found on small scales in phase-space and long time-scales. Chapter 2 presents a brief survey of the use of the idea of renormalization in physics. The first universal structure, described in Chapter 3, appears at the accumulation of period-doubling sequences, the conservative parallel to the Feigenbaum-Coullet-Tresser universality. The second one appears at the breakup of golden invariant circles, described and explained in Chapter 4. The renormalization picture is presented as the key to understanding the transition from regular to chaotic motion in area-preserving maps.