Let $f$ be a continuous self–map of a compact metric space $X$. The transformation $f$ induces in a natural way a self–map $\bar{f}$ defined on the hyperspace $K(X)$ of all nonempty closed subsets of $X$. We study which of the most usual notions of chaos for dynamical systems induced by $f$ are inherited by $\bar{f}$ and vice versa. We consider distributional chaos, Li–Yorke chaos, $\omega$-chaos, Devaney chaos, topological chaos (positive topological entropy), specification property and their variants. We answer questions stated independently by Roman-Flores and by Banks.