  We establish a direct link between Dunkl operators and quantum Lax matrices
$\mathcal L$ for the Calogero--Moser systems associated to an arbitrary Weyl
group $W$ (or an arbitrary finite reflection group in the rational case). This
interpretation also provides a companion matrix $\mathcal A$ so that $\mathcal
L, \mathcal A$ form a quantum Lax pair. Moreover, such an $\mathcal A$ can be
associated to any of the higher commuting quantum Hamiltonians of the system,
so we obtain a family of quantum Lax pairs. These Lax pairs can be of various
sizes, matching the sizes of orbits in the reflection representation of $W$,
and in the elliptic case they contain a spectral parameter. This way we
reproduce universal classical Lax pairs by D'Hoker-Phong and
Bordner-Corrigan-Sasaki, and complement them with quantum Lax pairs in all
cases (including the elliptic case, where they were not previously known). The
same method, with the Dunkl operators replaced by the Cherednik operators,
produces quantum Lax pairs for the generalised Ruijsenaars systems for
arbitrary root systems. As one of the main applications, we calculate a Lax
matrix for the elliptic $BC_n$ case with nine coupling constants (van Diejen
system), thus providing an answer to a long-standing open problem.
