Although batteries possess high energy storage density, their output power is limited by the slow movement of charge carriers, and thus capacitors are often required to deliver high power output. Dielectric capacitors have high power density with fast discharge rate, but their energy density is typically much lower than electrochemical supercapacitors. Increasing the energy density of dielectric materials is highly desired to extend their applications in many emerging power system applications. In this paper, we review the mechanisms and major characteristics of electric energy storage with electrochemical supercapacitors and dielectric capacitors. Three types of in-house-produced ferroic nonlinear dielectric thin film materials with high energy density are described, including (Pb(0.97)La(0.02))(Zr(0.90)Sn(0.05)Ti(0.05))O(3) (PLZST) antiferroelectric ceramic thin films, Pb(Zn(1/3)Nb(2/3))O(3-)Pb(Mg(1/3)Nb(2/3))O(3-)PbTiO(3) (PZN-PMN-PT) relaxor ferroelectric ceramic thin films, and poly(vinylidene fluoride) (PVDF)-based polymer blend thin films. The results showed that these thin film materials are promising for electric storage with outstandingly high power density and fairly high energy density, comparable with electrochemical supercapacitors.