This review covers resistive random access memories which utilize redox processes and ionic motion on the nanoscale as their storage principle (ReRAM). Generic aspects are described in order to provide the physics and chemistry background for the explanation of the microscopic switching mechanism and of the high nonlinearity in the switching kinetics. The valence change memory (VCM) effect is elaborated in more detail. As common features, ReRAM typically show very short switching times, low switching energies, and long data retention times. In addition, they offer a scalability potential down to feature sizes in the order of 5 nm and below.