The authors show that in a hereditarily indecomposable Banach space, briefly H.I. space, generators of \(C_0\)-groups and \(C_0\)-semigroups exhibit various very special properties. The generator A of a \(C_0\)-group on such a space is always bounded, and if the group has polynomial growth of order k there is a unique point \(\lambda_A\) in the spectrum \(\sigma\)(A) of A such that \((A - \lambda_A \text{Id})^k\) is compact. The generator A of a \(C_0\)-semigroup \((e^{tA})_{t\geq 0}\) on a H.I. space needs not be bounded. Nevertheless, the spectral mapping theorem \(\sigma(e^{tA})\backslash\{0\} = e^{t\sigma(A)}, t\geq 0,\) always holds. Finally, a detailed analysis of the structure of the spectrum of an (unbounded) operator, in particular of a generator, is given.