In this paper, we give some characterizations of generators of fractional resolvent families in terms of the resolvents of the generators and their derivatives. We first give a condition under which a densely defined operator on a Banach space can generate a bounded fractional resolvent family, which generalizes the result of Gomilko for $C_0$-semigroups. And this condition is also necessary in a Hilbert space. Another type of generation theorem for exponentially bounded fractional resolvent families is also presented.