As the penetration of variable renewable generation increases in power systems, issues, such as grid stiffness, larger frequency deviations, and grid stability, are becoming more relevant, particularly in view of 100% renewable energy networks, which is the future of smart grids. In this context, energy storage systems (ESSs) are proving to be indispensable for facilitating the integration of renewable energy sources (RESs), are being widely deployed in both microgrids and bulk power systems, and thus will be the hallmark of the clean electrical grids of the future. Hence, this article reviews several energy storage technologies that are rapidly evolving to address the RES integration challenge, particularly compressed air energy storage (CAES), flywheels, batteries, and thermal ESSs, and their modeling and applications in power grids. An overview of these ESSs is provided, focusing on new models and applications in microgrids and distribution and transmission grids for grid operation, markets, stability, and control.