I'll survey the history of how renormalisation ideas entered the study of breakup of invariant tori for Hamiltonian dynamics, including contributions by Greene, Chirikov, Escande & Doveil, Kadanoff & Shenker, myself, Aubry & Peyrard, Khanin & Sinai, Haydn, Wilbrink, Gallavotti & Benfatto, Stirnemann, Morrison, Jauslin and Koch, and describe the associated interpretation in terms of decimation of incommensurate structures for Frenkel-Kontorova (FK) chains. The result is a beautiful extension of KAM theory (which gives conditions for persistence of invariant tori for perturbations of integrable systems, equivalently sliding states for FK chains) to understand the boundary of the KAM domain and beyond (cantori, pinned states). Nevertheless, many of the consequences of the renormalisation view remain to be explored; some of the main questions will be highlighted.