The advent of acoustic metasurfaces (AMs), which are the two-dimensional equivalents of metamaterials, has opened up new possibilities in wave manipulation using acoustically thin structures. Through the interaction between the acoustic waves and the subwavelength scattering, AMs exhibit versatile capabilities to control acoustic wave propagation such as by steering, focusing, and absorption. In recent years, this vibrant field has expanded to include tunable, reconfigurable, and programmable control to further expand the capacity of AMs. This paper reviews recent developments in AMs and summarizes the fundamental approaches for achieving tunable control, namely, by mechanical tuning, active control, and the use of field-responsive materials. An overview of basic concepts in each category is first presented, followed by a discussion of their applications and details about their performance. The review concludes with the outlook for future directions in this exciting field.