Salim Tayou, speaking in the Institute for Advanced Study's Special Year Research Seminar, surveys what is known and conjectured about the modularity of special cycles in orthogonal and unitary Shimura varieties. He opens with the classical result, going back to Jacobi and Siegel, that theta series of quadratic lattices produce modular forms, then turns to the vast generalization proved by Kudla and Millson: generating series built from special cycles in these higher-dimensional varieties are themselves modular forms. Tayou lays out the conjectural framework that extends this picture further, points to recent progress on proving pieces of it, and frames the open problems that remain. Delivered at IAS's Simonyi Hall as the first of a multi-part series, the talk assumes graduate-level familiarity with Shimura varieties, lattices, and modular forms, and is aimed at researchers in arithmetic geometry and number theory rather than newcomers to the field.