The usual means of storing electrical energy are either batteries, where the current induces chemical reactions, or capacitors, where especially chosen dielectrics enhance the stored energy. Since capacitors can be discharged far more quickly than batteries and fuel cells, they have much higher power densities. At present, highly insulating polymers with large breakdown fields, such as polypropylene, are the dielectrics of choice. Nevertheless, their energy densities are quite low because of small dielectric constants. Ferroelectric polymers from the polyvinylidene fluoride (PVDF) family have significantly larger dielectric constants, yet their energy densities are still rather low. However, an admixture of a small amount of another polymer results in a dramatic (up to sevenfold) increase in the stored energy. We discuss the origin of this highly non-linear effect as due to a cooperative phase transition induced by the electric field, and describe the atomic transformations involved in this process. Fine-tuning of the kinetics of energy release may enable further dramatic advances in energy storage.