The focus of this talk will be on recent results on abstract homomorphisms between the groups of rational points of algebraic groups. I will indicate how these fit into the general framework of rigidity and also give a historical perspective, starting with results of Cartan and Schreier and van der Waerden. I will then discuss the main results of the landmark 1973 paper of Borel and Tits and address a conjecture on abstract homomorphisms that they formulated there. Our results settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring.