Let \(K\) be a global field of characteristic \(\neq 2\), \(S\) be a finite set of primes of \(K\) including all infinite primes, and let \({\mathcal O}_S(K)\) be the ring of algebraic \(S\)-integers in \(K\). In this paper, the structure of the Witt ring \(W{\mathcal O}_S(K)\) is determined. This Witt ring is finitely generated as an abelian group; its decomposition into cyclic summands is determined by explicitly describing generators of these summands given by \(1\)-dimensional, binary, or \(2\)-fold Pfister forms. In the case of a global number field and for \(S\) the set of infinite primes (i.e. \({\mathcal O}_S(K)\) is the ring of algebraic integers), this has already been done by the author in [Pr. Nauk. Uniw. Slask. Katowicach 1751, Ann. Math. Silesianae 12, 105-121 (1998; Zbl 0924.11026)], and the methods used there have been generalized in the present paper. The author also studies under which conditions there exist Witt ring isomorphisms \(W{\mathcal O}_S(K)\cong W{\mathcal O}_{S'}(L)\), and necessary and sufficient conditions for the existence of Witt ring isomorphisms \(WK\to WL\) mapping the Witt ring \(W{\mathcal O}_{S}(K)\) onto \(W{\mathcal O}_{S'}(L)\). The paper is complemented by various examples (including some for global function fields).