In this paper we begin the theory and application of the U(n+1) generalization of the classical Bailey Transform and Bailey Lemma. We work in the setting of multiple basic hypergeometric series very-well-poised on unitary groups U(n+1). The U(n+1) Bailey Transform is obtained from a suitably modified U(n+1) terminating very-well-poised 4 , 3 summation theorem and termwise transformations. It is then interpreted as a matrix inversion result for two infinite, lowertriangular matrices. This provides a higher-dimensional generalization of Andrews' matrix inversion formulation of the Bailey Transform. As in the classical case, the concept of a U(n+1) Bailey Pair is introduced, and then inverted. This U(n+1) inversion enables us to derive U(n+1) terminating balanced 3 , 2 summations directly from suitable U(n+1) terminating very-well-poised 6 , 5 summations, and vice-versa. These pairs of U(n+1) ``dual'' identities extend the classical one-variable case of Andrews. Special limiting cases of the U(n+1) terminating balanced 3 , 2 summation yield U(n+1) q-Gauss summations, several U(n+1) q-Chu-Vandermonde summations, U(n+1) q-binomial theorems, and a U(n+1) Cauchy identity. Many other consequences of the U(n+1) 6 , 5 and 3 , 2 summations, and the U(n+1) Bailey Transform are further developed in our subsequent papers. These include a derivation of the U(n+1) Bailey Lemma, several terminating U(n+1) q-Whipple transformations, U(n+1) 10 , 9 transformations, and nonterminating U(n+1) q-Whipple transformations. The classical case of all this work, corresponding to A 1 or equivalently U(2), contains a substantial amount of the theory and application of one-variable basic hypergeometric series. 1997 Academic Press 1. INTRODUCTION Jackson's [45] terminating balanced 3 , 2 summation [17, 35] and Rogers ' [74] terminating very-well-poised 6 , 5 summation [17,35] are two of the most important summation theorems in the theory and application of one-variable basic hypergeometric series [6,9,17,35,78]. The terminating article no.