  We summarize recent results on the construction of Lax pairs with spectral
parameter for the twisted and untwisted elliptic Calogero-Moser systems
associated with arbitrary simple Lie algebras, their scaling limits to Toda
systems, and their role in Seiberg-Witten theory. We extend part of this work
by presenting a new parametrization for the spectral curves for elliptic spin
Calogero-Moser systems associated with SL(N).
