- Chebyshev publishes On Primary Numbers in which he proves new results in the theory of prime numbers. He proves Bertrand's conjecture there is always at least one prime between and for .
- In his paper On a New Class of Theorems Sylvester first uses the word "matrix". (See this History Topic.)
- Bolzano's book Paradoxien des Undendlichen (Paradoxes of the Infinite) is published three years after his death. It introduces his ideas about infinite sets.
- Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers". In particular he gave the example 0.1100010000000000000000010000... where there is a 1 in place and 0 elsewhere.
- Riemann's doctoral thesis contains ideas of exceptional importance, for example "Riemann surfaces" and their properties.
- Hamilton publishes Lectures on Quaternions.
- Shanks gives π to 707 places (in 1944 it was discovered that Shanks was wrong from the 528th place).
- Riemann completes his Habilitation. In his dissertation he studied the representability of functions by trigonometric series. He gives the conditions for a function to have an integral, what we now call the condition of "Riemann integrability". In his lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854 he defines an -dimensional space and gives a definition of what today is called a "Riemannian space".
- Boole publishes The Laws of Thought on Which are founded the Mathematical Theories of Logic and Probabilities. He reduces logic to algebra and this algebra of logic is now known as Boolean algebra.
- Cayley makes an important advance in group theory when he makes the first attempt, which is not completely successful, to define an abstract group. (See this History Topic.)
- Maxwell publishes On Faraday's lines of force showing that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.
- Weierstrass publishes his theory of inversion of hyperelliptic integrals in Theorie der Abelschen Functionen which appeared in Crelle's Journal.
- Riemann publishes Theory of abelian functions. It develops further the idea of Riemann surfaces and their topological properties, examines multi-valued functions as single valued over a special "Riemann surface", and solves general inversion problems special cases of which had been solved by Abel and Jacobi.
- Cayley gives an abstract definition of a matrix, a term introduced by Sylvester in 1850, and in A Memoir on the Theory of Matrices he studies its properties.
- Möbius describes a strip of paper that has only one side and only one edge. Now known as the "Möbius strip", it has the surprising property that it remains in one piece when cut down the middle. Listing makes the same discovery in the same year.
- Dedekind discovers a rigorous method to define irrational numbers with "Dedekind cuts". The idea comes to him while he is thinking how to teach differential and integral calculus.
- Mannheim invents the first modern slide rule that has a "cursor" or "indicator".
- Riemann makes a conjecture about the zeta function which involves prime numbers. It is still not known whether Riemann's hypothesis is true in general although it is known to be true in millions of cases. It is perhaps the most famous unsolved problem in mathematics in the 21st century.