The memristor as a basic circuit element has drawn wide attention in the international research community of electrical engineers, physicists, and biologists. Memristors -being prominent for the realization of non-volatile VLSI resistive random access memories (RRAM) and for their applicability as synapses in neuromorphic systems provide even further surprising properties in electric circuits, which opens up new horizons to future applications. This article is in accordance with the special session of ISCAS2012 having the same title. We present a tutorial overview of the state-of-the-art of theory and applications of single memristor devices as well as memristive circuits.