- Hilbert poses 23 problems at the Second International Congress of Mathematicians in Paris as a challenge for the 20th century. The problems include the continuum hypothesis, the well ordering of the real numbers, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of "Dirichlet's principle" and many more. Many of the problems were solved during the 20th century, and each time one of the problems was solved it was a major event for mathematics.
- Goursat begins publication of Cours d'analyse mathematique which introduces many new analysis concepts.
- Fredholm develops his theory of integral equations in Sur une nouvelle méthode pour la résolution du problème de Dirichlet.
- Fejér publishes a fundamental summation theorem for Fourier series.
- Levi-Civita and Ricci-Curbastro publish Méthodes de calcul differential absolu et leures applications in which they set up the theory of tensors in the form that will be used in the general theory of relativity 15 years later.
- Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.
- Planck proposes quantum theory. (See this History Topic.)
- The Runge-Kutta method for numerically solving ordinary differential equations is proposed.
- Lebesgue formulates the theory of measure.
- Dickson publishes Linear groups with an exposition of the Galois field theory.
- Lebesgue gives the definition of the "Lebesgue integral".
- Beppo Levi states the axiom of choice for the first time.
- Gibbs publishes Elementary Principles of Statistical Mechanics which is a beautiful account putting the foundations of statistical mechanics on a firm foundation.
- Zermelo uses the axiom of choice to prove that every set can be well ordered.
- Lorentz introduces the "Lorentz transformations". (See this History Topic.)
- Poincaré proposes the Poincaré Conjecture, namely that any closed 3-dimensional manifold which is homotopy equivalent to the 3-sphere must be the 3-sphere.
- Poincaré gives a lecture in which he proposes a theory of relativity to explain the "Michelson and Morley experiment". (See this History Topic.)
- Einstein publishes the special theory of relativity. (See this History Topic.)
- Lasker proves the decomposition theorem for ideals into primary ideals in a polynomial ring.
- Fréchet, in his dissertation, investigated functionals on a metric space and formulated the abstract notion of compactness.
- Markov studies random processes that are subsequently known as "Markov chains".
- Bateman applies Laplace transforms to integral equations.
- Koch publishes Une methode geometrique elementaire pour l'etude de certaines questions de la theorie des courbes plane which contains the "Koch curve". It is a continuous curve which is of infinite length and nowhere differentiable.
- Fréchet discovers an integral representation theorem for functionals on the space of "quadratic Lebesgue integrable functions". A similar result was discovered independently by Riesz.
- Einstein publishes his principle of equivalence, in which says that gravitational acceleration is indistinguishable from acceleration caused by mechanical forces. It is a key ingredient of general relativity. (See this History Topic.)
- Heegaard and Dehn publish Analysis Situs which marks the beginnings of combinatorial topology.
- Brouwer's doctoral thesis on the foundations of mathematics attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School.
- Dehn formulates the word problem and the isomorphism problem for group presentations.
- Riesz proves the theorem now called the "Riesz-Fischer theorem" concerning Fourier analysis on Hilbert space.
- Gosset introduces "Student's -test" to handle small samples.
- Hardy and Weinberg present a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. This establishes the mathematical basis for population genetics.
- Zermelo publishes Untersuchungen über die Grundlagen der Mengenlehre (Investigations on the Foundations of Set Theory). He bases set theory on seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity. This aims to overcome the difficulties with set theory encountered by Cantor.
- Poincaré publishes Science et méthode (Science and Method), perhaps his most famous popular work.