For any root system Δ and a set of vectors which form a single orbit of the reflection (Weyl) group G_Δ generated by Δ, a spin Calogero–Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member μ of , to be called a 'site', we associate a vector space V_μ whose element is called a 'spin'. Its dynamical variables are the canonical coordinates of a particle in R^r (r = rank of Δ) and spin exchange operators (ρ Δ) which exchange the spins at the sites μ and s_ρ(μ). Here s_ρ is the reflection generated by ρ. For each Δ and a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by 'freezing' the canonical variables at the equilibrium point of the corresponding classical Calogero–Moser model. For Δ = A_r and = set of vector weights it reduces to the well-known Haldane–Shastry model. Universal Lax pair operators for both spin Calogero–Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.