Résumé : I will discuss recent progress in the analytic study of classical partition identities, including the famous « sum-product » formulas of Rogers-Ramanujan, Schur, and Capparelli. Such identities are rich in automorphic objects such as Jacobi theta functions, mock theta functions, and false theta functions. Furthermore, there are interesting connections to the combinatorics of multi-colored partitions, and the calculation of standard modules for Lie algebras and vertex operator theory.