With the anticipated increase in renewable wind and solar energy, efficient storage of electrical energy for balancing energy generation and demand on various time scales is becoming increasingly important. This review summarises the fundamentals of electrical energy storage in capacitors and batteries, discusses the similarities and differences of the two classes of devices and addresses the potential impact of the current research. A number of new materials and device structures that are anticipated to play roles in new devices are reviewed in greater detail. Topics covered are carbon nanotubes (CNTs), graphene, nanostructured transition metal oxides, silicon-based devices, ionic liquids (ILs), ion-conductive polymers and asymmetric supercapacitors.