Introduction. A 3-dimensional inversion geometry over an ordered field V in which every nonnegative number is a square may be defined as a partially ordered set II of objects called points, circles, spheres, and inversion space with the properties: (i) if p is any point, then there is an affine geometry whose "points," "lines," "planes," and "3-space" are, respectively, the points of II other than p, the circles containing p, the spheres containing p, and the inversion space; (ii) the underlying field of this affine geometry is V; (iii) this affine geometry can be made a Euclidean geometry in such a way that the "circles" and "spheres" of the Euclidean geometry are, respectively, the circles of II not containing p and the spheres of II not containing p. The purpose of this paper is to give axioms for II that will be sufficient to establish (i), (ii), and (iii). The only undefined relation is the ordering relation <, which means, geometrically, that all our axioms are incidence axioms. There does not seem to be any particular interest in finding alternative statements of (i), so (i) is simply assumed (1.4). Additional assumptions are added (2.11 and 2.12), and the remainder of the paper is devoted to proving that these axioms are sufficient for (ii) and (iii). The extension of this work to higher dimensions is straightforward, and we have concentrated on the 3-dimensional case for the sake of simplicity. The 2-dimensional case, however, is different in many ways('), and will be treated in a future paper. It is rather surprising that the literature contains so few investigations of the foundations of inversion geometry as an autonomous subject(2). Certainly much less is known about the postulates for inversion geometry than for other geometries. The present paper is an effort to remedy this deficiency. We wish to thank H. S. M. Coxeter, Tong Hing, and E. R. Lorch for their invaluable advice at various stages in the preparation of this manuscript. 1. The first set of postulates. In this section, we postulate that our set