TeX is one popular means by which to typeset complex mathematical formulae; it has been noted as one of the most sophisticated digital typographical systems in the world. TeX is popular in academia, especially in mathematics, computer science, engineering, and physics. It has largely displaced Unix troff, the other favored formatter, in many Unix installations, which use both for different purposes. It is now also being used for many other typesetting tasks, especially in the form of LaTeX and other template packages.

The widely-used MIME type for TeX is application/x-tex. TeX is free software.

When the first volume of Knuth's The Art of Computer Programming was published in 1969, it was typeset using hot metal type set by a Monotype Corporation typecaster with a hot metal typesetting machine from the 19th century which produced a "good classic style" appreciated by Knuth. When the second edition of the second volume was published, in 1976, the whole book had to be typeset again because the Monotype technology had been largely replaced by photographic techniques, and the original fonts were no longer available. However, when Knuth received the galley proofs of the new book on 30 March 1977, he found them awful. Around that time, Knuth saw for the first time the output of a high-quality digital typesetting system, and became interested in digital typography. The disappointing galley proofs gave him the final motivation to solve the problem at hand once and for all by designing his own typesetting system. On May 13, 1977, he wrote a memo to himself describing the basic features of TeX.

He planned to finish it on his sabbatical in 1978, but as it happened the language was frozen only in 1989, more than ten years later. Guy Steele happened to be at Stanford during the summer of 1978, when Knuth was developing his first version of TeX. When Steele returned to MIT that autumn, he rewrote TeX's I/O to run under the ITS operating system. The first version of TeX was written in the SAIL programming language to run on a PDP-10 under Stanford's WAITS operating system. For later versions of TeX, Knuth invented the concept of literate programming, a way of producing compilable source code and cross-linked documentation typeset in TeX from the same original file. The language used is called WEB and produces programs in DEC PDP-10 Pascal.

A new version of TeX, rewritten from scratch and called TeX82, was published in 1982. Among other changes, the original hyphenation algorithm was replaced by a new algorithm written by Frank Liang. TeX82 also uses fixed-point arithmetic instead of floating-point, to ensure reproducibility of the results across different computer hardware, and includes a real, Turing-complete programming language, following intense lobbying by Guy Steele.

In 1989, Donald Knuth released new versions of TeX and METAFONT. Despite his desire to keep the program stable, Knuth realised that 128 different characters for the text input were not enough to accommodate foreign languages; the main change in version 3.0 of TeX is thus the ability to work with 8-bits inputs, allowing 256 different characters in the text input.

Since version 3, TeX has used an idiosyncratic version numbering system, where updates have been indicated by adding an extra digit at the end of the decimal, so that the version number asymptotically approaches π. This is a reflection of the fact that TeX is now very stable, and only minor updates are anticipated. The current version of TeX is 3.1415926; it was last updated in March 2008. The design was frozen after version 3.0, and no new feature or fundamental change will be added, so all newer versions will contain only bug fixes. Even though Donald Knuth himself has suggested a few areas in which TeX could have been improved, he indicated that he firmly believes that having an unchanged system that will produce the same output now and in the future is more important than introducing new features. For this reason, he has stated that the "absolutely final change (to be made after my death)" will be to change the version number to π, at which point all remaining bugs will become features. Likewise, versions of METAFONT after 2.0 asymptotically approach e, and a similar change will be applied after Knuth's death.

However, since the source code of TeX is essentially in the public domain (see below), other programmers are allowed (and explicitly encouraged) to improve the system, but are required to use another name to distribute the modified TeX, meaning that the source code can still evolve. For example, the Omega project was developed after 1991, primarily to enhance TeX's multilingual typesetting abilities. Donald Knuth himself created "unofficial" modified versions, such as TeX-XeT, which allows a user to mix texts written in left-to-right and right-to-left writing systems in the same document.

TeX commands commonly start with a backslash and are grouped with curly braces. However, almost all of TeX's syntactic properties can be changed on the fly which makes TeX input hard to parse by anything but TeX itself. TeX is a macro and token-based language: many commands, including most user-defined ones, are expanded on the fly until only unexpandable tokens remain which get executed. Expansion itself is practically side-effect free. Tail recursion of macros takes no memory, and if-then-else constructs are available. This makes TeX a Turing-complete language even at expansion level.

The system can be divided into four levels: in the first, characters are read from the input file and assigned a category code (sometimes called "catcode", for short). Combinations of a backslash (really: any character of category zero) followed by letters (characters of category 11) or a single other character are replaced by a control sequence token. In this sense this stage is like lexical analysis, although it does not form numbers from digits. In the next stage, expandable control sequences (such as conditionals or defined macros) are replaced by their replacement text. The input for the third stage is then a stream of characters (including ones with special meaning) and unexpandable control sequences (typically assignments and visual commands). Here characters get assembled into a paragraph. TeX's paragraph breaking algorithm works by optimizing breakpoints over the whole paragraph. The fourth stage breaks the vertical list of lines and other material into pages.