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For α∈ (0, ∞), let H∞α (or H∞α,0) denote the collection of all functionsf which are analytic on the unit disc D and satisfy |f(z)|(1 -|z|2)α = O(1) (or|f(z)|(1 -|z|2)α = o(1) as |z| → 1). H∞α(or H∞α,0) is called a Bers-type space(or a little Bers-type space).In this paper, we give some basic properties of H∞α.Cψ,the composition operator associated with a symbol function ψ which is an analyticself map of D, is difined by Cψf = f o ψ. We characterize the boundedness andcompactness of Cψ which sends one Bers-type space to another function space.