- M Suzuki discovers new infinite families of finite simple groups.
- Edward Lorenz discovers a simple mathematical system with chaotic behaviour. It leads to the new mathematics of chaos theory which is widely applicable.
- Smale proves the higher dimensional Poincaré conjecture for , namely that any closed -dimensional manifold which is homotopy equivalent to the -sphere must be the -sphere.
- Jacobson publishes his classic text Lie algebras.
- Sobolev publishes Applications of Functional Analysis in Mathematical Physics.
- John Thompson and Feit publish Solvability of Groups of Odd Order which proves that all nonabelian finite simple groups are of even order. Their paper requires 250 pages to prove the theorem.
- Cohen proves the independence of the axiom of choice and of the continuum hypothesis.
- Hironaka solves a major problem concerning the resolution of singularities on an algebraic variety.
- Sergi Novikov's work on differential topology, in particular in calculating stable homotopy groups and classifying smooth simply-connected manifolds, leads him to make the "Novikov Conjecture".
- Bombieri uses his improved large sieve method to prove what is now called "Bombieri's mean value theorem", which concerns the distribution of primes in arithmetic progressions.
- Tukey and Cooley publish a paper introducing the "Fast Fourier Transform" algorithm.
- Selten publishes important work on distinguishing between reasonable and unreasonable decisions in predicting the outcome of games. It will lead to the award of a Nobel Prize in 1994.
- Grothendieck receives a Fields Medal for his work on geometry, number theory, topology and complex analysis. His theory of schemes allows certain of Weil's number theory conjectures to be solved. His theory of topoi is highly relevant to mathematical logic, he had given an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve.
- Lander and Parkin use a computer to find a counterexample to Euler's Conjecture. They find .
- Alan Baker proves "Gelfond's Conjecture" about the linear independence of algebraic numbers over the rational numbers.
- Atiyah publishes K-theory which details his work on -theory and the index theorem which led to the award of a Fields Medal in 1966.
- Novikov and Adian jointly publish a proof that the Burnside group is infinite for every and every .