a classical theorem of jordan asserts that every finite subgroup $b$ of the complex general linear (matrix) group $gl(n)$ contains an abelian (normal) subgroup $a$ such that the index of $a$ in $b$ does not exceed a universal constant that depends only on $n$. we discuss analogues of jordan's theorem where the matrix group is replaced by the group of biregular (or birational) automorphisms of a complex algebraic variety, or by the diffeomorphism group of a smooth real manifold.