Dense sphere packings

How to place equal balls in the $N$-dimensional Euclidean space in the densest possible way? The problem is non-trivial even in dimension $N=2$. In dimension 3 the answer was obtained at the very end of 20th century, and for $N=4$ it is yet unknown.
In the introduction I shall present the history, the statement of the problem, and some well-known results. The main part is devoted to astounding results of Maryna Viazovska, who solved the problem in dimensions 8 and 24 using quasimodular forms. At the end I plan to recall some results of mine for very large dimensions $(N \to \infty)$, based on algebraic geometry and number theory.