  What is the minimum number of triangles in a graph of given order and size?
Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first
non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s
in 1955; it is now known as the Erd\H{o}s-Rademacher problem. After attracting
much attention, it was solved asymptotically in a major breakthrough by
Razborov in 2008. In this paper, we provide an exact solution for all large
graphs whose edge density is bounded away from~$1$, which in this range
confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we
give a description of the extremal graphs.
