As the location-based service (LBS) plays an increasingly important role in real life, the topic of positioning attracts more and more attention. Under different environments and principles, researchers have proposed a series of positioning schemes and implemented many positioning systems. With widely deployed networks and massive devices, wireless fidelity (Wi-Fi) technology is promising in the field of indoor positioning. In this paper, we survey the authoritative or latest positioning schemes for Wi-Fi-assisted indoor positioning. To this end, we describe the problem and corresponding applications, as well as an overview of the alternative methods. Then, we classify and analyze Wi-Fi-assisted indoor positioning schemes in detail, as well as review related work. Furthermore, we point out open challenges and forecast promising directions for future work.