There has been a great deal of work done in recent years on weighted Bergman spaces A p α on the unit ball B n of C n , where 0 < p < ∞ and α > −1. We extend this study in a very natural way to the case where α is any real number and 0 < p ≤ ∞. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space H 2 , and the so-called Arveson space. Some theorems in the paper are relatively simple generalizations of the classical case α >

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... −1, while others require substantial additional effort. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.