Acoustic metasurfaces derive their characteristics from the interaction between acoustic waves and specifically designed materials. The field is driven by the desire to control acoustic wave propagation using compact devices and is governed by fundamental and physical principles that provide the design rules and the functionality of a wave. Acoustic metasurfaces have added value and unusual functionalities compared with their predecessor in materials science, namely, acoustic metamaterials. These rationally designed 2D materials of subwavelength thickness provide a new route for sound wave manipulation. In this Review, we delineate the fundamental physics of metasurfaces, describe their different concepts and design strategies, and discuss their functionalities for controllable reflection, transmission and extraordinary absorption. In particular, we outline the main designs of acoustic metasurfaces, including those based on coiling-up space, Helmholtz-resonator-like and membrane-type structures, and discuss their applications, such as beam focusing, asymmetrical transmission and self-bending beams. We conclude with an outlook of the future directions in this emerging field.