The Conformable Battery encompasses a class of devices wherein the outer casing of the battery may be configured in a variety of shapes to fit specific applications including flat planar designs (circular, square, rectangular) and three-dimensional geometric shapes (e.g., curved plates, thin cylinders with hollow cores, other geometric shapes). The internal structure of the battery consists of multi-cell compartments where the walls of the compartments are bonded to the outer skin contributing to the batteries' structural stiffness, integrity, and ability to sustain moderate to high internal pressure, with enhanced ability to dissipate heat. The design is applicable to the patented Bimodal Battery concept (U.S. Pat. No. 6,187,471B1) as well as other battery design concepts.