With increasing concerns about climate change, there is a transition from high-carbon-emitting fuels to green energy resources in various applications including household, commercial, transportation, and electric grid applications. Even though renewable energy resources are receiving traction for being carbon-neutral, their availability is intermittent. To address this issue to achieve extensive application, the integration of energy storage systems in conjunction with these resources is becoming a recommended practice. Additionally, in the transportation sector, the increased demand for EVs requires the development of energy storage systems that can deliver energy for rigorous driving cycles, with lithium-ion-based batteries emerging as the superior choice for energy storage due to their high power and energy densities, length of their life cycle, low self-discharge rates, and reasonable cost. As a result, battery energy storage systems (BESSs) are becoming a primary energy storage system. The high-performance demand on these BESS can have severe negative effects on their internal operations such as heating and catching on fire when operating in overcharge or undercharge states. Reduced efficiency and poor charge storage result in the battery operating at higher temperatures. To mitigate early battery degradation, battery management systems (BMSs) have been devised to enhance battery life and ensure normal operation under safe operating conditions. Some BMSs are capable of determining precise state estimations to ensure safe battery operation and reduce hazards. Precise estimation of battery health is computed by evaluating several metrics and is a central factor in effective battery management systems. In this scenario, the accurate estimation of the health indicators (HIs) of the battery becomes even more important within the framework of a BMS. This paper provides a comprehensive review and discussion of battery management systems and different health indicators for BESSs, with suitable classification based on key characteristics.