We give a complete characterization of the sequences β = (β n) of positive numbers for which all composition operators on H 2 (β) are bounded, where H 2 (β) is the space of analytic functions f on the unit disk D such that ∞ n=0 |a n | 2 β n < +∞ if f (z) = ∞ n=0 a n z n. We prove that all composition operators are bounded on H 2 (β) if and only if β is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on H 2 (β) if and only if β is slowly oscillating. We give applications of our results.