Rapid charge and discharge rates have become an important feature of electrical energy storage devices, but cause dramatic reductions in the energy that can be stored or delivered by most rechargeable batteries (their energy capacity). Supercapacitors do not suffer from this problem, but are restricted to much lower stored energy per mass (energy density) than batteries. A storage technology that combines the rate performance of supercapacitors with the energy density of batteries would significantly advance portable and distributed power technology. Here, we demonstrate very large battery charge and discharge rates with minimal capacity loss by using cathodes made from a self-assembled three-dimensional bicontinuous nanoarchitecture consisting of an electrolytically active material sandwiched between rapid ion and electron transport pathways. Rates of up to 400C and 1,000C for lithium-ion and nickel-metal hydride chemistries, respectively, are achieved (where a 1C rate represents a one-hour complete charge or discharge), enabling fabrication of a lithium-ion battery that can be 90% charged in 2 minutes.