During the past twenty years extensive work has been done on the ring C(X). The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X. A significant part of this work has recently been summarized in the book [2]. Concerning the ring C(X, Z), very little has been written. This is natural, since C(X, Z) is less important in problems of topology and analysis than C(X). Nevertheless, for some problems of topology, analysis and algebra, C(X, Z) is a useful tool. Moreover, a comparison of the theories of C(X) and C(X, Z) should illuminate those aspects of the theory of C(X) which derive from the special properties of the field of real numbers. For these reasons it seems worthwhile to devote some attention to C(X, Z). The paper is divided into six sections. The first of these treats topological questions. An analogue of the Stone-Cech compactification is developed and studied. In ?2, the ideals in C(X, Z) are related to the filters in a certain lattice of sets. The correspondence is similar to that which exists between the ideals of C(X) and the filters in the lattice of zero sets of continuous functions on X. This theory provides a characterization of those ideals of C(X, Z) which are intersections of maximal ideals. ?3 is concerned with the space of maximal ideals in C(X, Z). In ?4, some existence theorems for maximal ideals are proved. The residue class fields of C(X, Z) modulo maximal ideals are studied in the last two sections. It turns out that those of prime characteristic are trivial: the integers modulo the characteristic. The residue class fields of characteristic zero are distinctly nontrivial. In ?5, the cardinality of such fields is investigated. The main result is that they are always uncountable. In ?6, the algebraic properties of the zero characteristic residue class fields are examined. It is shown for example that these fields are always quasialgebraically closed. As we noted above, very little has been published concerning the ring C(X, Z). Nevertheless, a considerable number of "folk theorems" exist in the subject. One of our objectives in writing this paper is to get these results