Extending the concept of hypercyclic semigroups, e.g., studied in [W. Desch, W. Schappacher and G. F. Webb, Ergodic Theory Dyn. Syst. 17, 793–819 (1997; Zbl 0910.47033)], the authors give conditions such that the solution of a nonautonomous abstract Cauchy problem \(\dot{x}(t)=A(t)x(t)\), \(t\geq 0\), forms a dense subset of the underlying Banach space.