  A new formulation of Calogero-Moser models based on root systems and their
Weyl group is presented. The general construction of the Lax pairs applicable
to all models based on the simply-laced algebras (ADE) are given for two types
which we call `root' and `minimal'. The root type Lax pair is new; the matrices
used in its construction bear a resemblance to the adjoint representation of
the associated Lie algebra, and exist for all models, but they do not contain
elements associated with the zero weights corresponding to the Cartan
subalgebra. The root type provides a simple method of constructing sufficiently
many number of conserved quantities for all models, including the one based on
$E_{8}$, whose integrability had been an unsolved problem for more than twenty
years. The minimal types provide a unified description of all known examples of
Calogero-Moser Lax pairs and add some more. In both cases, the root type and
the minimal type, the formulation works for all of the four choices of
potentials: the rational, trigonometric, hyperbolic and elliptic.
