In his last letter to G. H. Hardy, Ramanujan wrote about his so-called "mock theta functions". These functions are now largely understood (by work of Zwegers) in the context of nonholomorphic Jacobi forms. Here we discuss recent work which pertains to the roles of these mock theta functions in Zagier's theory of quantum modular forms, and we construct new mock theta functions (one for each rational elliptic curve) from the Weierstrass zeta-function and we explain their relevance in the study of the arithmetic of elliptic curves.