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Polymers play a pivotal role in advancing energy storage technologies because of their unique properties, including high conductivity, flexibility, and environmental stability. The integration of artificial intelligence (AI) and machine learning (ML) into polymer science has revolutionized the design, discovery, and optimization of polymeric materials, enabling the development of faster, cost-effective, and innovative solutions. This review explores the transformative impact of AI and ML in polymer science, focusing on their applications in polymer design, characterization, and energy storage. Key advancements include the accelerated discovery of materials, predictive modeling of polymer properties, and high-throughput screening of polymer candidates. This review highlights the strengths of AI and ML, including their ability to handle complex datasets, optimize multiproperty trade-offs, and uncover hidden relationships between structure and properties. However, challenges such as data limitations, model interpretability, and synthetic feasibility remain significant barriers to progress. The paper also identifies gaps in the literature, including the need for improved structural descriptors, expanded datasets, and the integration of physical principles into ML models. Future directions emphasize the development of sustainable polymers, generative design frameworks, and interdisciplinary collaboration to address pressing global challenges. By leveraging AI and ML, this work aims to accelerate the development of next-generation polymers for energy storage, fostering clean, efficient, and sustainable energy solutions.