- Alan Baker is awarded a Fields Medal for his work on Diophantine equations.
- Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.
- Stephen Cook formulates the versus problem regarding polynomial time algorithms.
- Thom publishes Structural Stability and Morphogenesis which explains catastrophe theory. The theory examines situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes, and has important applications in biology and optics.
- Quillen formulates higher algebraic -theory, a new tool that uses geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory.
- Deligne proves the three "Weil conjectures".
- Chen Jingrun shows that every sufficiently large even integer is the sum of a prime and a number with at most two prime factors. It makes a major contribution to the Goldbach Conjecture.
- Feigenbaum discovers a new constant, approximately 4.669201609102..., which is related to period-doubling bifurcations and plays an important part in chaos theory.
- Mandelbrot publishes Les objets fractals, forme, hasard et dimension which describes the theory of fractals.
- Lakatos work Proofs and Refutations is published as a book two years after his death. First published in four parts in 1963-64 the work gives Lakatos's account of how mathematics develops.
- Thurston is awarded the Oswald Veblen Geometry Prize of the American Mathematical Society for his work on foliations.
- Appel and Haken show that the Four Colour Conjecture is true using 1200 hours of computer time to examine around 1500 configurations. (See this History Topic.)
- Adleman, Rivest, and Shamir introduce public-key codes, a system for passing secret messages using large primes and a key which can be published.
- Connes publishes work on non-commutative integration theory.