The orthogonal polynomials of my title are orthogonal polynomials in several variables ; there is (at least) one family of them associated with each root system, and they possess the symmetry of the corresponding Weyl group. They depend on two or more parameters, and for certain limiting values of these parameters, they can be recognized as zonal spherical functions on semisimple Lie groups over $\bf R$ and over ${\bf Q}_p$ ; thus they form a bridge between harmonic analysis on real Lie groups and on their $p$-adic counterparts. Some years ago the author showed how to construct these polynomials, and formulated various conjectures about them. These conjectures have recently been proved in full generality by Cherednik using the theory of affine Hecke algebras.