Abstract. We study a partial order on the regions of R" dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introduction. Let % -{77,, H2,.. .,Hk] be a set of hyperplanes in R". Then the components of R" -UHe.xH form a set 9t of open n-cells we will call regions. Traditionally 9t has been studied in terms of enumeration, for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in %. For a thorough discussion of this problem see Zaslavsky's monograph [Zl] and the references therein. See also [OS and Te].In this paper we will study 9? from a different perspective. Our concern will be the way in which one can move from one region to another by "walking through" one hyperplane at a time. To analyze this situation we define a partial order on 9t relative to a base region where the cover relations correspond to " walking through" a hyperplane and moving further away from the base.The paper is organized as follows: In §1 we define the partial order on 3? and begin to study its structure. This culminates in Theorem 1.11 where an explicit computation of its Mobius function is performed. In §11 we examine the special case where all the hyperplanes go through the origin. In certain cases we are able to compute the homotopy type of these posets. The next three sections are devoted to applications. §111 applies our work to a partial order on a zonotope. §IV deals with the weak Bruhat order of a Weyl group as a special case of our posets. As a corollary we show that the homotopy type of the weak Bruhat order is that of a sphere.Finally, §V applies our results to a partial order on the acyclic orientations of a graph. Some of the relationships used in § §III, IV and V appear in an enumerative context in the excellent article of Greene and Zaslavsky [GZ].Most of our poset notation is standard and can be found in [A]. When more than one partial order is being used we will sometimes use subscripts to indicate where the relation is, i.e., x ^Py means that x is less than or equal toy in the poset P. By A(.P) we mean the simplicial complex of chains (totally ordered sets) in P. We will think of A(7") as its geometric realization. By P* we mean the order dual of P, i.e.,