Energy storage can find a wide application in electric power systems, both at the grid and at the customer level. A great contribution can be brought by storage, in terms of system reliability, integration of renewable sources, power quality and effective economic exploitation of energy resources, especially in the framework of a deregulated market. Several energy storage technologies exist or are under development, each offering specific performances in terms of, energy and power capacity, response time, affordable number of charge/discharge cycles and efficiency. Two types of storage technology rely on superconductivity namely, superconducting magnetic energy storage (SMES), and flywheel energy storage system (FESS) based on superconducting magnetic bearings (SMB). These technologies are able to provide a remarkable round trip efficiency in the range 85-95 %. Moreover they are also able to provide a fast response time and can afford a large number of charge/discharge cycles. These are demanding requirements difficult to reach with alternative storage methods. SMES and superconducting flywheels are discussed in detail in the following, along with their application to the power system. In particular a general description of the specifics of an energy storage system is given in section 2. In section 3 the possible applications of energy storage are reviewed and the specific requirements needed for each application are pointed out. The main characteristics of SMES and SMB-based FESS technologies are resumed in section 4 and 5 respectively. The relevant issues related to the main components, including the power conditioning system, are reviewed. A short overview of the state of the art for both the technologies is also provided.