  We show how the elliptic Calogero-Moser integrable systems arise from a
symplectic quotient construction, generalising the construction for A_{N-1} by
Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted)
affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a
natural geometric construction of Lax operators for these systems. We elaborate
on the connection of the associated Hamiltonians to superpotentials for N=1*
deformations of N=4 supersymmetric gauge theory, and argue how non-perturbative
physics generates the elliptic superpotentials. We also discuss the relevance
of these systems and the associated quotient construction to open problems in
string theory. In an appendix, we use the theory of orbit algebras to show the
systematics behind the folding procedures for these integrable models.
