The fastest growth in electricity demand in the industrialized world will likely come from the broad adoption of artificial intelligence (AI) - accelerated by the rise of generative AI models such as OpenAI’s ChatGPT. The global “data center arms race” is driving up power demand and grid stress, which creates local and regional challenges because people in the area understand that the additional data center-related electricity demand is coming from faraway places, and they will have to support the additional infrastructure while not directly benefiting from it. So, there is an incentive for the data center operators to manage the fast and unpredictable power surges internally so that their loads appear like a constant baseload to the electricity grid. Such high-intensity and short-duration loads - which can have very high tariffs - can be served by Hybrid Energy Storage Systems (HESS) combining multiple types of storage technologies. Each storage type can respond to different timescales of load variations, enabling fast power balancing and reduced battery degradation. Properly designed HESS architectures, and their optimal operations, will make AI data center loads as baseloads, which will help the data center operators avoid high-tariff electricity. This review paper presents an overview of current energy storage technologies, their classifications, and recent performance data, focusing on their suitability for AI-driven computing. Technical requirements such as fast response, modularity, and long cycle life - that are critical for sustainable and reliable data center operations - are discussed. Special attention is given to Northern Virginia, the world’s largest data center hub, where rapid load growth and grid limitations highlight the urgent need for hybrid storage deployment. The paper concludes by identifying key challenges, research gaps, and future directions for integrating hybrid energy storage with on-site generation to optimally manage the high variability in the data center load and build sustainable, low-carbon, and intelligent AI data centers.