- Drinfeld is awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan for his work on quantum groups and for his work in number theory.
- Menasco and Thistlethwaite prove the knot theory conjecture known as "Tait's Second Conjecture", namely that any two reduced alternating diagrams of the same prime knot are related by a sequence of twists.
- A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation has no solutions for and coprime integers .
- Borcherds is awarded a Fields Medal for his work in automorphic forms and mathematical physics; Gowers receives one for his work in functional analysis and combinatorics; Kontsevich receives one for his work in algebraic geometry, algebraic topology, and mathematical physics; and McMullen receives one for his work on holomorphic dynamics and geometry of 3-dimensional manifolds.
- Thomas Hales proves Kepler's problem on sphere packing.
- The Great Internet Mersenne Prime Search project finds the 38th Mersenne prime: 26972593 -1.
- Conrad and Taylor prove the "Taniyama-Shimura conjecture". Wiles proved a special case in 1993 on his way to giving a proof of Fermat's Last Theorem.