Let ℳ n be a linear hyperplane arrangement in ℝ n . We define two corresponding posets G k (ℳ n and V k (ℳ n ) of oriented matroids, which approximate the Grassmannian G k (ℝ n ) and the Stiefel manifold V k (ℝ n ). The basic conjectures are that the "OM-Grassmannian"G k (ℳ n ) has the homotopy type of G k (ℝ n ), and that the "OM-Stiefel bundle" Δπ: ΔV k (ℳ n ) → ΔG k (ℳ n ) is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids ℳ n . © 1993 Springer-Verlag New York Inc.