Jordan Groups and Geometric Properties of Manifolds

The aim of this note is to draw attention to recent results about the so called Jordan property of groups. (The name was motivated by a classical theorem of Jordan about finite subgroups of matrix groups). We explore interrelations between geometric properties of complex projective varieties and compact Kähler manifolds and the Jordan property (or the lack of it) of their automorphism groups of birational and biregular selfmaps, and of bimeromorphic and biholomorphic maps, respectively.