The study of cycles on homogeneous varietes has seen a great deal of activity in the past few years. In particular, new results about cycles on quadric hypersurfaces has resulted in fundamental breakthroughs in the theory of quadratic forms. The goal of this talk will be to answer a basic question about cycles on homogeneous of more general types. In particular, we will address the problem of calculating the group of zero dimensional cycles on such varieties. In the case of quadrics, this was first done by Swan in 1989 (and also independently by Karpenko). Similar computations were made later for certain classes of other homogeneous varieties by Merkurjev and Panin. By using the geometry of Hilbert schemes of points on homogeneous varieties, we will describe how to extend the previous results and to compute the group of zero cycles for some homogeneous varieties of each of the classical types.