Automorphism group is an important invariant of a smooth projective variety or more generally a compact Kahler manifold X. I will show that if the dimension of X is at least 3 and the nef cone of X satisfies several specific conditions (C) then X has no “interesting” automorphism. A simple heuristic argument shows that these conditions (C) are expected to hold on a “generic” compact Kahler manifold. I will also discuss the (uni)rationality of some quotients E^3/G (here E is an elliptic curve and G is a finite group). Smooth models of these examples both solve a question posed by Kenji Ueno in 1975 and give first explicit examples of smooth rational 3-folds with interesting automorphisms. The proof that these manifolds have interesting automorphisms makes use of a tool called dynamical degrees, which are bimeromorphic invariants of pairs (X: compact Kahler manifold, f:X->X dominant meromorphic). The talk will end with some open questions.