The first half of the talk will concern automorphisms of compact quantum groups, which can be thought of as "inner" automorphisms, in the group theoretic sense. The behaviour of Normal subgroups of compact quantum groups will be studied under these automorphisms. In the second half we will study discrete groups acting on compact quantum groups acting as quantum group automorphisms, we will see that ergodicity here implies the notion of topological transitivity. We will end by studying actions of integers, and a certain question of automorphisms of Tarski Monsters, which will show how the quantum case can be very different from the classical one.