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:

Attenuation curves exhibit a wide range of UV-through-optical slopes, from curves with shallow (Milky Way-like) slopes to those exceeding the slope of the Small Magellanic Cloud extinction curve.

The slopes of the curves correlate strongly with the effective optical opacities, in the sense that galaxies with lower dust column density (lower visual attenuation) tend to have steeper slopes, whereas the galaxies with higher dust column density have shallower (grayer) slopes.

Galaxies exhibit a range of 2175- Alpha UV bump strengths, including no bump, but, on average, are suppressed compared with the average MilkyWay extinction curve.

Theoretical studies indicate that both the correlation between the slope and the dust column as well as variations in bump strength may result from geometric and radiative transfer effects.