- Solving several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in the 1800s invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.
- Unifying the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers").
- Providing the first correct and complete axiomatization of Euclidean geometry to replace Euclid's axiomatization of geometry, in his 1899 book Grundlagen der Geometrie ("Foundations of Geometry").
- His suggestion in 1920 that mathematics be formulated on a solid logical foundation (by showing that all of mathematics follows from a system of axioms, and that that axiom system is consistent). This is still the most popular philosophy of mathematics usually called formalism. Unfortunately, Gödel's Incompleteness Theorem showed that his grand plan was impossible.
- Hilbert's paradox of the Grand Hotel, a musing about strange properties of the infinite.
- Laying the foundations of functional analysis by studying integral equations and Hilbert spaces.
- Putting forth an influential list of 23 unsolved problems in the Paris conference of the International Congress of Mathematicians in 1900.
- Hilbert helped provide the basis for the Theory of Automata which was later built upon by computer scientist Alan Turing.
- Jeremy Gray, 2000, The Hilbert Challenge, ISBN 0198506511