A zone diagram is a new variation of the classical notion of the Voronoi diagram. Given points (sites) p(1),..., p(n) in the plane, each pi is assigned a region R(i), but in contrast to the ordinary Voronoi diagrams, the union of the R(i) has a nonempty complement, the neutral zone. The de. ning property is that each R(i) consists of all x is an element of R(2) that lie closer (nonstrictly) to pi than to the union of all the other R(j) j not equal i. Thus, the zone diagram is defined implicitly, by a "fixed-point property," and neither its existence nor its uniqueness seem obvious. We establish existence using a general fixed-point result (a consequence of Schauder's theorem or Kakutani's theorem); this proof should generalize easily to related settings, say higher dimensions. Then we prove uniqueness of the zone diagram, as well as convergence of a natural iterative algorithm for computing it, by a geometric argument, which also relies on a result for the case of two sites in an earlier paper. Many challenging questions remain open.