Hexagonal boron nitride nanosheets (BNNSs) are two-dimensional nanomaterials with graphitic-like layered nanostructures, high surface areas, and large aspect ratios. Owing to their excellent thermal conductivity, electrical and mechanical strengths, BNNSs are emerging as multifunctional fillers in polymer dielectrics. In this article, the authors review the recent progress in the BN-containing polymer nanocomposites designed for high-performance film capacitors. While general synthetic approaches to BNNSs and polymer/BNNS nanocomposites are summarized, particular attention is placed on structure-property correlation and rational structural design of the composites with optimized dielectric properties and capacitive performances. In stark contrast to the polymer composites employing high dielectric constant fillers to enhance the electric displacement, a new design concept based on the utilization of BNNSs with a wide bandgap to impede electrical conduction and consequently improve breakdown strength and charge-discharge efficiency of the polymer composites, is highlighted. The significance of developing dielectric capacitors with desirable thermal conductivity and thermal stability to ensure their robust and efficient operation is emphasized. The merits and challenges regarding the existing polymer dielectrics containing BNNSs for energy storage are identified. An outlook for future research opportunities and engineering applications is also presented in this review.