Chronology

Dedekind publishes his formal construction of real numbers and gives a rigorous definition of an integer.
Heine publishes a paper which contains the theorem now known as the "Heine-Borel theorem".
Société Mathématique de France is founded.
Méray publishes Nouveau précis d'analyse infinitésimale which aims to present the theory of functions of a complex variable using power series.
Sylow publishes Théorèmes sur les groupes de substitutions which contains the famous three "Sylow theorems" about finite groups. He proves them for permutation groups.
Klein gives his inaugural address at Erlanger. He defines geometry as the study of the properties of a space that are invariant under a given group of transformations. This became known as the "Erlanger programm" and profoundly influences mathematical development.

Maxwell publishes Electricity and Magnetism. This work contains the four partial differential equations, now known as "Maxwell's equations".
Hermite publishes Sur la fonction exponentielle (On the Exponential Function) in which he proves that $e$ is a transcendental number.
Gibbs publishes two important papers on diagrams in thermodynamics.
Brocard produces his work on the triangle.

Cantor publishes his first paper on set theory. He rigorously describes the notion of infinity. He shows that infinities come in different sizes. He proves the controversial result that almost all numbers are transcendental.

Gibbs publishes On the Equilibrium of Heterogeneous Substances which represents a major application of mathematics to chemistry.

Cantor is surprised at his own discovery that there is a one-one correspondence between points on the interval [0, 1] and points in a square.

Kempe published his false proof of the Four Colour Theorem. (See this History Topic.)
Lexis publishes On the theory of the stability of statistical series which begins the study of time series.
Kharkov Mathematical Society is founded.