I will discuss results of three different types in geometry and topology.(1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras.(2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels.(3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.