Abstract: Mobile application compatibility failure refers to a kind of software defect caused by application running results inconsistent with expected results in different environments or internal changes. Due to the highly open-source nature of the Android platform, mobile application compatibility failures frequently occur on the Android platform. In the most serious cases, this kind of failure can lead to program crashes. On the one hand, it will affect the user experience. On the other hand, its sudden occurrence will bring huge losses to users. The combinations of different device models and versions of the Android OS make it impossible for developers to test their applications thoroughly. In the context of the extremely serious fragmentation of the Android ecosystem, how to effectively deal with the compatibility faults has become a hot research issue in software quality assurance. Starting from the three aspects of Android mobile application compatibility fault (i.e., fault analysis, fault detection, fault location, and repair), we briefly introduce the development history of Android mobile application compatibility test and the main challenges faced in this field. In addition, we also review and summarize the practical exploration and theoretical achievements in recent years. Finally, we discuss future work, which can provide guidelines for other researchers to study compatibility testing for Android mobile application.