The approximate mechanism design for facility location problems lies in the intersection of optimization and computational economics. It has been extensively studied in the last decades, largely due to its practical importance in various domains, such as social planning and clustering. At a high level, the goal is to design mechanisms to select a set of locations on which to build a set of facilities, aiming to optimize some social objective and ensure desirable properties based on the preferences of strategic agents, who might have incentives to misreport their private information such as their locations. This chapter introduces the significant progress that has been made since the introduction of this problem, highlighting the different variants and methodologies, as well as some interesting future directions.