In this chapter we study to what extent the exponential type of certain orbits of a C0—semigroup is determined by boundedness properties of the resolvent in a given right half-plane. The prototype of such results has already been proved in Section 2.2, where we showed that ω0(T) = s0(A) for C0—semigroups on Hilbert spaces. We shall prove a generalization of this result for semigroups in arbitrary Banach spaces.