I will report on new and recent work with Leandro Arosio (Universita di Roma, Tor Vergata) on chaos and other aspects of dynamics in certain highly symmetric complex manifolds. For example, we prove that for many linear algebraic groups G, chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of G. I will start with some background on holomorphic dynamics in general, on chaos, and on the highly symmetric complex manifolds to which our results apply.