First, the impedance source idea is used to alleviate the problems of Voltage Source Inverters (VSI), and Current Source Inverters (CSI), which leads to the introduction of the Z-Source Inverter (ZSI). Afterwards, not only did the Z-source cells were used in the inverter topologies, but they were also added to other power electronic converters, such as dc-dc choppers, ac-dc rectifiers, and ac-ac converters. Moreover, these power electronic converters, which have Z-source cells, are widely used in different applications. In particular, the existing single-phase ac-ac Z-source converters are reviewed in this paper, and their different features are studied in detail. In the past decades, the single-phase Z-source ac-ac converters have attracted significant attention due to their unique features, such as single-stage power conversion, boost in-phase, and buck out-of-phase features. Therefore, several modified or improved structures along with different control methods have been introduced to solve their problems and limitations. Based on the four features, these structures can be categorized into four groups: without magnetic coupling, with magnetic coupling, able to change the frequency, and inherent commutation. In this review, a general diagram is depicted, which many of the existing topologies can be obtained from it. Additionally, new topologies can be obtained by having this general diagram. After studying these structures, a comparative study is made to obtain the advantages and disadvantages of each. In the end, some advice and recommendations are given to develop and obtain new structures.