Abstract: A classical theorem of Jordan asserts that every finite subgroup $B$ of the complex general linear (matrix) group $GL(n)$ contains a (normal) commutative subgroup $A$ such that the index of $A$ in $B$ does not exceed a universal constant that depends only on $n$. One may ask whether analogues of Jordan's theorem remain true if the matrix group is replaced by the group of biregular (or birational) automorphisms of a complex algebraic variety. We discuss results of J.-P. Serre, V. L. Popov, Yu. Prokhorov--K. Shramov, T. Bandman and the speaker.