  This paper explains the recent developments on the symplectic theory of
Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or
not. One fundamental ingredient of these developments has been the
understanding of singular affine structures. These developments make use of
results obtained by many authors in the second half of the twentieth century,
notably Arnold, Duistermaat and Eliasson, of which we also give a concise
survey. As a motivation, we present a collection of remarkable results proven
in the early and mid 1980s in the theory of Hamiltonian Lie group actions by
Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many
people, including the authors, to work on more general Hamiltonian systems. The
paper concludes discussing a spectral conjecture for quantum integrable
systems.
