Some time ago, John Millson and I studied the locally symmetric Riemannian cycles -- special cycles -- on arithmetic quotients for classical groups G = O(p,q), U(p,q), Sp(p,q). One of our main results was that the generating function for the cohomology classes determined by certain of these cycles is in fact a modular form for a group Sp(r), U(r,r), O*(2r). In certain cases, in particular for U(p,q), our cycles include algebraic cycles. In this talk, I will describe the results of a current joint project with Michael Rapoport in which we study the arithmetic version of the special cycles in the case of the Shimura variety for U(1,n-1).