Human computation systems (HCSs) have been widely adopted in various domains. Their goal is to harness human intelligence to solve computational problems that are beyond the capability of modern computers. One of the most challenging problems in HCSs is how to incentivize a broad range of users to participate in the system and make high efforts. This article surveys the field of HCSs from the perspective of incentives and mechanism design. We first review state-of-the-art HCSs, focusing on how incentives are provided to users. We then use mechanism design to theoretically analyze different incentives. We survey the mechanisms derived from state-of-the-art HCSs as well as classic mechanisms that have been used in HCSs. Finally, we discuss eight promising research directions for designing incentives in HCSs.