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Let D be the unit disc and H(D) be the set of all analytic functions on D. In [2], C. Cowen defined a space H = {f ∈ H(D): f(z) = ∞∑k=0 ak(z + 1)k, (A)z ∈ D, ||f||2 =∞∑k=0|ak|24k ＜∞}. In this article, the authors consider the similar Hardy spaces with arbitrary weights and discuss some properties of them. Boundedness and compactness of composition operators between such spaces are also studied.