Complexity in geometry and in physics is a key feature of many materials, devices, and networks. The richness of the resulting phenomena can enable new capabilities of great importance for the production, storage, and transmission of energy, which are critical issues for sustainability of our planet and the human population. On the other hand, the complexity greatly complicates design and operation, so that mathematics and simulation can be essential for understanding, designing, and characterizing these systems. Examples presented here include the following: For batteries, increased performance requirements involve more complex physics, including phase transitions and nonequilibrium thermodynamics. The goal for solar cells is to increase their efficiency by fabricating systems with large internal surface areas but high conductivity. The electrical power grid is a complex network for which there is a great need for early detection of anomalies that could lead to failure. Research on each of these examples involves significant interaction of mathematics with science and engineering. In addition, the speakers will describe the industrial significance of their results. Their presentation in a AAAS symposium promises to stimulate further interdisciplinary research on energy issues.