Next-generation energy storage materials aim to combine the best characteristics of batteries and supercapacitors, producing devices that concurrently deliver high energy and power densities. In order to achieve this goal, the kinetics of ion and electron transport within the electrodes must be enhanced while maintaining a large volume fraction of electrolytically active material for energy storage. In this regard, the idealized electrode microstructure has been envisioned as a three-dimensional, co-continuous arrangement of percolating domains that allow for large interfacial contact between the constituent phases and low-resistance paths for ion and electron transport. In this dissertation, I report a novel method to produce this unique microstructure through the use of soft matter templates derived from bicontinuous interfacially jammed emulsion gels (bijels). These non-equilibrium soft materials inherently have a co-continuous microstructure, where two interpenetrating fluid domains are kinetically arrested through jamming of particles at the fluid-fluid interface during spinodal decomposition. The neutrally-wetting colloids adsorb to the fluid interface upon phase separation and become jammed when the interfacial area is sufficiently reduced to just accommodate them. The characteristic domain size, which establishes both the pore diameter and the internal surface area, is controlled solely through the overall colloid volume fraction and can be tuned over a wide range. Through simple chemical post processing steps, bijels are converted to electrolytically active composites. Combined, the morphological control afforded during bijel formation and our chemical processing steps allow for independent tailoring of the microstructural parameters that govern the electrode’s electrochemical performance. In this work, I present composite electrodes that I have synthesized through this route and demonstrate how their salient electrochemical characteristics are dictated by these morphological parameters, allowing for concurrent delivery of high energy and power, bridging the gap between modern batteries and supercapacitors.