What are great examples of comprehensively archived mathematical correspondence (including both handwritten and electronic items)?
Context: polished papers usually don't reveal the full process that accompanied the way definitions and proofs were conceived. For example, Colmez's introduction to the book Courbes et fibrés vectoriels en théorie de Hodge p-adique by Fargues and Fontaine, contains long excerpts of email exchanges which are very interesting.
The Euler archive contains more or less everything Leonard Euler ever wrote that still exists today, including his letter correspondence with various people.
Gauss' correspondence is also easily available, for example with Wolfgang Bolyai, including the (in)famous passages about the discovery of non-Euclidean geometry.
Kurt Godel's Collected Works include his correspondence in volumes IV and V.
The end of Solomon Feferman's preface to volume III is an amusing historical document in itself, from 1995 -- I see it as capturing the brief moment when one could perceive e-mail as an alternative to physical correspondence, rather than a default:
"One feature of our work together [on the editorial team] in recent years deserves special mention: namely, the use of the electronic mail system, which has radically transformed and facilitated our communications and decision-making, and has produced a new art of nagging. How did we ever live without it?"
One unusual example is Serge Lang, The File: Case Study in Correction, 1977-1979 (Springer-Verlag, 1981), with both correspondence and the published documents which provoked it. Lang is the central correspondent in the file, and Neal Koblitz and Saunders Mac Lane also have large enough roles to appear in the table of contents.
"The File is a collection of documents from a major dispute involving a number of American college professors, mainly mathematicians, statisticians, and sociologists. The controversy was ignited by the mathematician Serge Lang's reaction to a questionnaire, 'The 1977 Survey of the American Professoriate', distributed by E. C. Ladd of the University of Connecticut and S. M. Lipset of Stanford. The ensuing discussion - in part acrimonious and personal - soon involved a large group of active and passive participants, and included issues such as survey techniques, evaluation of academic work, public and political honesty, and McCarthyism at Harvard."
The Grothendieck-Mumford correspondence
These letters have been published in Volume II of Mumford's complete works (Selected Papers II. On Algebraic Geometry, Including Correspondence with Grothendieck)
2020 email correspondence between professor Władysław Narkiewicz, a very patient mathematician, and Dr. Kumar Eswaran, who claimed to have proven the Riemann conjecture.
It ends with this beautiful conclusion:
Although our views on your result are somewhat different I want you to know that I enjoyed our discussion which showed that the notion of a proof may have different interpretations.
A summary of the correspondence can be found at skeptics.stackexchange.com: Have mathematicians concluded that an Indian mathematical physicist has solved the Riemann Hypothesis?
The book "Der Briefwechsel Richard Dedekind – Heinrich Weber" transcribes the full correspondence between the two. From the description:
This volume provides the very first transcription of correspondence between Richard Dedekind and Heinrich Weber, one of the most important instances of written dialog between mathematicians in the 19th century. Nearly every subarea of mathematics is addressed in the letters, which intensively discuss nascent developments in the field. A register of persons and index of works ease access to the topics discussed in the letters.
Riposte armonie (hidden harmonies) is a book documenting the correspondence between Federigo Enriques and Guido Castelnuovo, over a span of 20 years. The book is unidirectional - it only contains letters written by Enriques - but it documents the golden years of the Italian school of algebraic geometry, as well as the friendship between the two mathematicians
The Grothendieck-Brown correspondence
As far as I know these letters are planned to be published alongside "Pursuing stacks" on Documents Mathématiques
Some years ago, I was tracking the origins of fractional calculus and finally found a comprehensive list of Leibniz correspondence.
This page https://agrothendieck.github.io/ is collecting (mainly from the https://grothendieck.umontpellier.fr/archives-grothendieck/) mathematical letters from Grothendieck (it is work in progress but now over 200 pg long):
https://agrothendieck.github.io/divers/letters.pdf
See "Réflexions Mathématiques" there for more or contact the owner for comments.
I am surprised that nobody has mentioned
Emil Artin und Helmut Hasse, Die Korrespondenz 1923--1934
Helmut Hasse und Emmy Noether, Die Korrespondenz 1925--1935
and
I'm not sure whether this is a comprehensive example, but
Singularités Irrégulières: Correspondance et Documents (Société Mathématique de France, 2007)
contains correspondence from 1976 - 1991 between Pierre Deligne, Bernard Malgrange, and Jean-Pierre Ramis on the subject of irregular singularities of linear differential equations.