Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation.

It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description:

TCS covers a wide variety of topics including algorithms, data structures, computational complexity, parallel and distributed computation, probabilistic computation, quantum computation, automata theory, information theory, cryptography, program semantics and verification, algorithmic game theory, machine learning, computational biology, computational economics, computational geometry, and computational number theory and algebra. Work in this field is often distinguished by its emphasis on mathematical technique and rigor.

While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved.

Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. In 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist practically relevant problems that are NP-complete – a landmark result in computational complexity theory.

Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below:

An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.

An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.