Abstract: The online core monitoring system (CMS) is a crucial system for ensuring the safe operation of nuclear power plants. Its primary function is to real-time monitor the key parameter changes within the reactor core, in order to timely detect potential abnormal conditions and provide subsequent operational recommendations. The mathematical modeling of the main functions of online CMS was conducted, and its core modeling forward problem was clearly defined as well as the inverse problems of monitoring, diagnosis, and operational control. Through analyzing the research status of typical online CMS systems, the solution approaches for the aforementioned forward and inverse problems were identified, providing references for resolving specific engineering issues of online CMS in the future. The reactor core modeling mainly involves high and low-fidelity mechanistic models, as well as various machine learning models and model order reduction techniques. Solving the detector placement, physical field reconstruction, and uncertainty quantification problems using diverse types of detectors and data assimilation algorithms is the core of the monitoring inverse problem. The diagnosis inverse problem primarily relies on in-core and ex-core neutron detector information to analyze the causes of control rod drop, self-powered neutron detector anomalies, and vibration abnormalities. Utilizing the theoretical models of core tracking, various types of reactivity or power distribution control constitute the main tasks of the operational support inverse problem. Furthermore, this paper points out the existing issues in current online CMS technology research, such as multi-source information fusion, uncertainty quantification, and operational procedures. Particularly, considering the influence of various uncertainty factors within the framework of digital twins and numerical reactors, to enhance the plant operating capability, is one of the future development directions for online CMS and its underlying technologies.