- Donaldson publishes Self-dual connections and the topology of smooth 4-manifolds which leads to totally new ideas concerning the geometry of 4-manifolds.
- Faltings proves the "Mordell conjecture". He makes a major contribution to Fermat's Last Theorem showing that for every there are at most a finite number of coprime integers satisfying . (See THIS LINK.)
- Margulis proves the "Oppenheim conjecture" on the values of indefinite irrational quadratic forms at integer points.
- Zelmanov proves an important conjecture about when an infinite dimensional Lie algebra is nilpotent.
- Langlands is the first recipient of the National Academy of Sciences Award in Mathematics. He receives it for "extraordinary vision that has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory."
- Elkies finds a counterexample to Euler's Conjecture with , namely 26824404 + 153656394 + 187967604 = 206156734 .
Later in the year Frye finds the smallest counter-example: 958004 + 2175194 + 4145604 = 4224814 .
- Bourgain, using analytic and probabilistic methods, solves the problem which had been a longstanding one in "Banach space" theory and harmonic analysis.