Over the last fifteen years or so, there has emerged a satisfactory and coherenttheoryoforthogonalpolynomialsinseveralvariables,attachedtorootsystems,and depending on two or more parameters. At the present stage of its develop-ment, it appears that an appropriate framework for its study is provided by thenotionofanaffinerootsystem:toeachirreducibleaffinerootsystem S thereareassociated several families of orthogonal polynomials (denoted by E λ , P λ , Q λ ,P (ε)λin this book). For example, when S is the non-reduced affine root systemof rank 1 denoted here by (C ∨1 ,C 1 ), the polynomials P λ are the Askey-Wilsonpolynomials [A2] which, as is well-known, include as special or limiting casesall the classical families of orthogonal polynomials in one variable.I have surveyed elsewhere [M8] the various antecedents of this theory: sym-metric functions, especially Schur functions and their generalizations such aszonal polynomials and Hall-Littlewood functions [M6]; zonal spherical func-tions on p-adic Lie groups [M1]; the Jacobi polynomials of Heckman andOpdam attached to root systems [H2]; and the constant term conjectures ofDyson, Andrews et al. ([D1], [A1], [M4], [M10]). The lectures of Kirillov [K2]also provide valuable background and form an excellent introduction to thesubject.Thetitleofthismonographisthesameasthatofthelecture[M7].Thatreport,for obvious reasons of time and space, gave only a cursory and incompleteoverview of the theory. The modest aim of the present volume is to fill in thegaps in that report and to provide a unified foundation for the theory in itspresent state.The decision to treat all affine root systems, reduced or not, simultaneouslyonthesamefootinghasresultedinanunavoidablycomplexsystemofnotation.In order to formulate results uniformly it is necessary to associate to each affineroot system S another affine root system S ? (which may or may not coincidewith S), and to each labelling ( § 1.5) of S a dual labelling of S ? .