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Abstract

In this work, we begin with a survey of composition operators on the Hardy space $H^2$ and on the Wiener algebra $A^+$ of absolutely convergent Taylor series, with special emphasis on their compactness, or invertibility, or isometric character. The main results are due respectively to J. Shapiro and D.~Newman. In a second part, we present more recent results, due to Gordon and Hedenmalm on the one hand, and to Bayart, the author et al. on the other hand, concerning the analogues of $H^2$ and $A^+$ in the setting of Dirichlet series. We are led to the intermediate study of Taylor series in several, or countably many, variables. We finish with some open problems.