  The differential geometry of almost Grassmann structures defined on a
differentiable manifold of dimension n = pq by a fibration of Segre cones SC
(p, q) is studied. The peculiarities in the structure of almost Grassmann
structures for the cases p=q=2; p = 2, q > 2 (or p > 2, q = 2), and p > 2, q >
2 are clarified. The fundamental geometric objects of these structures up to
fourth order are derived. The conditions under which an almost Grassmann
structure is locally flat or locally semiflat are found for all cases indicated
above.
