For any analytic self-map φ of {z : |z| \u3c 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cφ to be closed-range on the Bloch space B . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cφ is closed-range on the Bergman space A2 , then it is closed-range on B , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem
