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Abstract: Let and be uniform algebras with Choquet boundaries δA and δB. A map is called norm-linear if ; norm-additive, if , and norm-additive in modulus, if for each and all algebra elements f and g. We show that for any norm-linear surjection there exists a homeomorphism such that for every and . Sufficient conditions for norm-additive and norm-linear surjections, not assumed a priori to be linear, or continuous, to be unital isometric algebra isomorphisms are given. We prove that any unital norm-linear surjection T for which , or which preserves the peripheral spectra of -peaking functions of A, is a unital isometric algebra isomorphism. In particular, we show that if a linear operator between two uniform algebras, which is surjective and norm-preserving, is unital, or preserves the peripheral spectra of -peaking functions, then it is automatically multiplicative and, in fact, an algebra isomorphism. [Copyright &y& Elsevier]