Due to its development agility, continuous delivery, scalability and other characteristics, the microservice architecture systems (MASs) have provided complex business functions to hundreds of millions of users in many application fields. The operation and maintenance governance for a large number of microservices with complex relationships is crucial to ensuring the stability and reliability of an MAS. Although this research field has received certain attention and produced some innovative results, there is a lack of systematic reviews covering the different aspects of it. In this context, the central objective of this study is to carry out a systematic literature review (SLR) in this field, in an attempt to review existing issues, discuss the main trends, and share the findings with the academia. As a result, we start from more than 500 scientific papers published from 2009 to 2021 and extract 144 most significant papers, identify that the main research directions of this field include load balancing, fault detection, and autoscaling. Subsequently, we provide a comprehensive description of these research directions, discuss them in particular detail. We also determine limitations of current work and discuss new directions worth exploring in the future. Consequently, the outcomes will assist professionals and experts in the industry as well as academic researchers to focus more on operation and maintenance governance of MASs and further improve the relevant methods and theoretical systems in this field.