A classical theorem of Jordan asserts that each finite subgroup of the complex general linear group GL(n) 

is ``almost commutative": it contains a commutative normal subgroup 

with index bounded by an universal constant that depends only on n.

We discuss an analogue of this property for the groups of birational (and biregular)

 automorphisms of complex algebraic varieties

and the groups of diffeomorphisms of real manifolds.