This volume and its companion volume (see SMFDM/13) reproduce, with notes and comments, the correspondence between Jean-Pierre Serre and John Tate from 1956 to 2000. They also contain a selection of their email correspondence after 2000. The texts are reproduced in their original language: in English or in French. Most of them are from 1956-1976. They treat questions such as the write-up of Bourbaki's Elements, Galois cohomology, rigid geometry, Tate's conjectures on algebraic cycles, formal and $p$-divisible groups, complex multiplication, and modular forms: congruence properties, weight 1 forms, and Galois representations. These volumes should be useful to people interested in number theory or the history of mathematics.