When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment of Volume 4 appeared in paperback as Fascicle 2 in 2005.
The hardback Volume 4A, combining Volume 4, Fascicles 0–4, was published in 2011. Volume 4, Fascicle 6 ("Satisfiability") was released in December 2015; Volume 4, Fascicle 5 ("Mathematical Preliminaries Redux; Backtracking; Dancing Links") was released in November 2019.
Volume 4B consists of material evolved from Fascicles 5 and 6.  The manuscript was sent to the publisher on August 1, 2022 and the volume was published in September 2022.
Fascicle 7, planned for Volume 4C, was the subject of Knuth's talk on August 3, 2022.
In January 1962, when he was a graduate student in the mathematics department at Caltech, Knuth was approached by Addison-Wesley to write a book about compiler design, and he proposed a larger scope. He came up with a list of twelve chapter titles the same day. In the summer of 1962 he worked on a FORTRAN compiler for UNIVAC. During this time, he also came up with a mathematical analysis of linear probing, which convinced him to present the material with a quantitative approach. After receiving his Ph.D. in June 1963, he began working on his manuscript, of which he finished his first draft in June 1965, at 3000 hand-written pages. He had assumed that about five hand-written pages would translate into one printed page, but his publisher said instead that about 1+1⁄2 hand-written pages translated to one printed page. This meant he had approximately 2000 printed pages of material, which closely matches the size of the first three published volumes. At this point, Knuth received support from Richard S. Varga, who was the scientific adviser to the publisher. Varga was visiting Olga Taussky-Todd and John Todd at Caltech. With Varga's enthusiastic endorsement, the publisher accepted Knuth's expanded plans. In its expanded version, the book would be published in seven volumes, each with just one or two chapters. Due to the growth in Chapter 7, which was fewer than 100 pages of the 1965 manuscript, per Vol. 4A p. vi, the plan for Volume 4 has since expanded to include Volumes 4A, 4B, 4C, 4D, and possibly more.
In 1976, Knuth prepared a second edition of Volume 2, requiring it to be typeset again, but the style of type used in the first edition (called hot type) was no longer available. In 1977, he decided to spend some time creating something more suitable. Eight years later, he returned with TEX, which is currently used for all volumes.
The offer of a so-called Knuth reward check worth "one hexadecimal dollar" (100HEXbase 16 cents, in decimal, is $2.56) for any errors found, and the correction of these errors in subsequent printings, has contributed to the highly polished and still-authoritative nature of the work, long after its first publication. Another characteristic of the volumes is the variation in the difficulty of the exercises. Knuth even has a numerical difficulty scale for rating those exercises, varying from 0 to 50, where 0 is trivial, and 50 is an open question in contemporary research.
All examples in the books use a language called "MIX assembly language", which runs on the hypothetical MIX computer. Currently,[when?] the MIX computer is being replaced by the MMIX computer, which is a RISC version. Software such as GNU MDK exists to provide emulation of the MIX architecture. Knuth considers the use of assembly language necessary for the speed and memory usage of algorithms to be judged.
Knuth was awarded the 1974 Turing Award "for his major contributions to the analysis of algorithms […], and in particular for his contributions to the 'art of computer programming' through his well-known books in a continuous series by this title."American Scientist has included this work among "100 or so Books that shaped a Century of Science", referring to the twentieth century,and within the computer science community it is regarded as the first and still the best comprehensive treatment of its subject.[failed verification] Covers of the third edition of Volume 1 quote Bill Gates as saying, "If you think you're a really good programmer… read (Knuth's) Art of Computer Programming… You should definitely send me a résumé if you can read the whole thing."The New York Times referred to it as "the profession's defining treatise".
22.214.171.124. Estimating backtrack costs (chapter 6 of "Selected Papers on Analysis of Algorithms", and Fascicle 5, pp. 44−47, under the heading "Running time estimates")
7.2.3. Generating inequivalent patterns (includes discussion of Pólya enumeration theorem) (see "Techniques for Isomorph Rejection", chapter 4 of "Classification Algorithms for Codes and Designs" by Kaski and Östergård)
Volume 3: Sorting and Searching. Second Edition (Reading, Massachusetts: Addison-Wesley, 1998), xiv+780pp.+foldout. ISBN 978-0-201-89685-5, 0-201-89685-0. Errata:  (2011-01-08),  (2020-03-26, 27th printing). Addenda:  (2011).
Volume 4A: Combinatorial Algorithms, Part 1. First Edition (Upper Saddle River, New Jersey: Addison-Wesley, 2011), xv+883pp. ISBN 978-0-201-03804-0, 0-201-03804-8. Errata:  (2020-03-26?, 22nd printing).
Volume 4B: Combinatorial Algorithms, Part 2. First Edition (Upper Saddle River, New Jersey: Addison-Wesley, 2023), xviii+714pp. ISBN 978-0-201-03806-4, 0-201-03806-4(2022-11-??, 2nd printing).
Volume 1, Fascicle 1: MMIX – A RISC Computer for the New Millennium. (Addison-Wesley, 2005-02-14) ISBN 0-201-85392-2. Errata:  (2020-03-16) (will be in the fourth edition of volume 1)
These volumes were superseded by newer editions and are in order by date.
Volume 1: Fundamental Algorithms. First edition, 1968, xxi+634pp, ISBN 0-201-03801-3.
Volume 2: Seminumerical Algorithms. First edition, 1969, xi+624pp, ISBN 0-201-03802-1.
Volume 3: Sorting and Searching. First edition, 1973, xi+723pp+foldout, ISBN 0-201-03803-X. Errata: .
Volume 1: Fundamental Algorithms. Second edition, 1973, xxi+634pp, ISBN 0-201-03809-9. Errata: .
Volume 2: Seminumerical Algorithms. Second edition, 1981, xiii+ 688pp, ISBN 0-201-03822-6. Errata: .
The Art of Computer Programming, Volumes 1-3 Boxed Set. Second Edition (Reading, Massachusetts: Addison-Wesley, 1998), pp. ISBN 978-0-201-48541-7, 0-201-48541-9
The Art of Computer Programming, Volumes 1-4A Boxed Set. Third Edition (Reading, Massachusetts: Addison-Wesley, 2011), 3168pp. ISBN 978-0-321-75104-1, 0-321-75104-3
Volume 4'sfascicles 0–4 were revised and published as Volume 4A:
Volume 4, Fascicle 0: Introduction to Combinatorial Algorithms and Boolean Functions. (Addison-Wesley Professional, 2008-04-28) vi+240pp, ISBN 0-321-53496-4. Errata:  (2011-01-01).
^"Reflections on a year of reading Knuth". infinitepartitions.com. Retrieved 2020-07-25. I worked, or at least attempted to work, every single problem in the first volume. In some cases I settled for just understanding what the question was trying to ask for. In some cases I failed even to accomplish that (don't judge me until you try it yourself). Each problem is assigned a difficulty rating from 0-50 where 0 is trivial and 50 is "unsolved research problem" (in the first edition, Fermat's last theorem was listed as a 50, but since Andrew Wiles proved it, it's bumped down to a 45 in the current edition). I think I was able to solve most of the problems rated < 20 — it was hit and miss beyond that. Most of the problems fell into the 20-30 difficulty range, but Knuth's idea of "difficult" is subjective, and problems that he considers to be of average difficulty ended up stretching my comparatively tiny brain painfully. I've never climbed Mount Everest, but I imagine the whole ordeal feels similar: painful while you're going through it, but triumphant when you reach the pinnacle.
^"Donald E. Knuth – A. M. Turing Award Winner". AM Turing. Retrieved 2017-01-25.
^Morrison, Philip; Morrison, Phylis (November–December 1999). "100 or so Books that shaped a Century of Science". American Scientist. Sigma Xi, The Scientific Research Society. 87 (6). Archived from the original on 2008-08-20. Retrieved 2008-01-11.
^Weinberger, Matt. "Bill Gates once said 'definitely send me a résumé' if you finish this fiendishly difficult book". Business Insider. Retrieved 2016-06-13.
^Lohr, Steve (2001-12-17). "Frances E. Holberton, 84, Early Computer Programmer". The New York Times. Retrieved 2010-05-17.
^ abWells, Mark B. (1973). "Review: The Art of Computer Programming, Volume 1. Fundamental Algorithms and Volume 2. Seminumerical Algorithms by Donald E. Knuth" (PDF). Bulletin of the American Mathematical Society. 79 (3): 501–509. doi:10.1090/s0002-9904-1973-13173-8.
Slater, Robert (1987). Portraits in Silicon. MIT Press. ISBN 0-262-19262-4.
Shasha, Dennis; Lazere, Cathy (1995). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Copernicus. ISBN 0-387-97992-1.
Overview of topics (Knuth's personal homepage)
Oral history interview with Donald E. Knuth at Charles Babbage Institute, University of Minnesota, Minneapolis. Knuth discusses software patenting, structured programming, collaboration and his development of TeX. The oral history discusses the writing of The Art of Computer Programming.
"Robert W Floyd, In Memoriam", by Donald E. Knuth - (on the influence of Bob Floyd)
TAoCP and its Influence of Computer Science (Softpanorama)