High power electrical energy storage systems are becoming critical devices for advanced energy storage technology. This is true in part due to their high rate capabilities and moderate energy densities which allow them to capture power efficiently from evanescent, renewable energy sources. High power systems include both electrochemical capacitors and electrostatic capacitors. These devices have fast charging and discharging rates, supplying energy within seconds or less. Recent research has focused on increasing power and energy density of the devices using advanced materials and novel architectural design. An increase in understanding of structure-property relationships in nanomaterials and interfaces and the ability to control nanostructures precisely has led to an immense improvement in the performance characteristics of these devices. In this review, we discuss the recent advances for both electrochemical and electrostatic capacitors as high power electrical energy storage systems, and propose directions and challenges for the future. We asses the opportunities in nanostructure-based high power electrical energy storage devices and include electrochemical and electrostatic capacitors for their potential to open the door to a new regime of power energy.