Over the last decade, the number of cyber attacks targeting power systems and causing physical and economic damages has increased rapidly. Among them, false data injection attacks (FDIAs) are a class of cyber-attacks against power grid monitoring systems. Adversaries can successfully perform FDIAs to manipulate the power system state estimation (SE) by compromising sensors or modifying system data. SE is an essential process performed by the energy management system towards estimating unknown state variables based on system redundant measurements and network topology. SE routines include bad data detection algorithms to eliminate errors from the acquired measurements, e.g. in case of sensor failures. FDIAs can bypass BDD modules to inject malicious data vectors into a subset of measurements without being detected, and thus manipulate the results of the SE process. To overcome the limitations of traditional residual-based BDD approaches, data-driven solutions based on machine learning algorithms have been widely adopted for detecting malicious manipulation of sensor data due to their fast execution times and accurate results. This study provides a comprehensive review of the most up-to-date machine learning methods for detecting FDIAs against power system SE algorithms.