His works include developments in algebra: He also wrote on probability and the quadrature of the circle.

He was a professor of mathematics at the University of Modena and a medical doctor[5] including scientific work[6] on typhus.

In 1799 Ruffini marked a major improvement for group theory, developing Joseph-Louis Lagrange's work on permutation theory ("Réflexions sur la théorie algébrique des équations", 1770–1771).

Lagrange's work was largely ignored until Ruffini established strong connections between permutations and the solvability of algebraic equations.

Ruffini was the first to assert, controversially, the unsolvability by radicals of algebraic equations higher than quartics, which angered many members of the community such as Gian Francesco Malfatti (1731–1807).