Summary: Let

$R$ and

$S$ be two irreducible root systems spanning the same vector space and having the same Weyl group

$W$, such that

$S$ (but not necessarily

$R$) is reduced. For each such pair

$(R,S)$ we construct a family of

$W$-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters

$q,{t}_{1},{t}_{2},\cdots ,{t}_{r}$, where

$r$ $(=1,2$ or 3) is the number of

$W$-orbits in

$R$. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and

$p$-adic symmetric spaces. Also when

$R=S$ is of type

${A}_{n}$, they conincide with the symmetric polynomials [described in

*I. G. Macdonald*, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press (1995;

Zbl 0824.05059), Chapter VI].