One-parameter families of unbounded linear operators acting in a Banach space X, and satisfying the semigroup and strong continuity properties on a suitable subspace of X, are discussed; the notion of infinitesimal generator is generalized to this unbounded setting, and a HilleYosida-type theorem is proved. The theory is illustrated by several examples, which include fractional integrals and derivatives acting in LP(O, 0o).