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.