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Understanding the properties of dust attenuation curves in galaxies and the physical mechanisms that shape them are among the fundamental questions of extragalactic astrophysics, with great practical significance for deriving the physical properties of galaxies. Attenuation curves result from a combination of dust grain properties, dust content, and the spatial arrangement of dust and different populations of stars. In this review, we assess the state of the field, paying particular attention to extinction curves as the building blocks of attenuation laws. We introduce a quantitative framework to characterize extinction and attenuation curves, present a theoretical foundation for interpreting empirical results, overview an array of observational methods, and review observational results at low and high redshifts. Our main conclusions include the following:
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