  While the intersection of the Grassmannian Bruhat decompositions for all
coordinate flags is an intractable mess, the intersection of only the {\em
cyclic shifts} of one Bruhat decomposition turns out to have many of the good
properties of the Bruhat and Richardson decompositions. This decomposition
coincides with the projection of the Richardson stratification of the flag
manifold, studied by Lusztig, Rietsch, and Brown-Goodearl-Yakimov. However, its
cyclic-invariance is hidden in this description. Postnikov gave many
cyclic-invariant ways to index the strata, and we give a new one, by a subset
of the affine Weyl group we call {\em bounded juggling patterns}. We adopt his
terminology and call the strata {\em positroid varieties.} We show that
positroid varieties are normal and Cohen-Macaulay, and are defined as schemes
by the vanishing of Plucker coordinates. We compute their T-equivariant Hilbert
series, and show that their associated cohomology classes are represented by
affine Stanley functions. This latter fact lets us connect Postnikov's and
Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus. Our principal
tools are the Frobenius splitting results for Richardson varieties as developed
by Brion, Lakshmibai, and Littelmann, and the Hodge-Grobner degeneration of the
Grassmannian. We show that each positroid variety degenerates to the projective
Stanley-Reisner scheme of a shellable ball.
