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f-divergences form a class of measures of distance between probability distributions; they are widely used in areas such as information theory and signal processing. In this paper, we unveil a new connection between f-divergences and differential privacy, a confidentiality policy that provides strong privacy guarantees for private data-mining; specifically, we observe that the notion of α-distance used to characterize approximate differential privacy is an instance of the family of f-divergences. Building on this observation, we generalize to arbitrary f-divergences the sequential composition theorem of differential privacy. Then, we propose a relational program logic to prove upper bounds for the f-divergence between two probabilistic programs. Our results allow us to revisit the foundations of differential privacy under a new light, and to pave the way for applications that use different instances of f-divergences.