In 2008 researchers at Hewlett-Packard announced the physical realization of the “memristor” which was theoretically predicted as a fundamental non-linear circuit element by Leon Chua in 1971. Since that time there have been numerous scientific papers applying the concept of memory resistors to a wide range of thin film materials used for a new type of non-volatile memory called ReRAM (resistive random-access memory). It has also been noted that memory resistors may be applicable to explain behavior of biological neurons and some research groups have developed circuit designs exploiting memory resistors as components of neuromorphic electronics. This article reviews the historical background of various forms of memory resistors including the Widrow-Hoff memistor, Chua’s memristor model, and a mem-resistor model I developed to correct some of the deficiencies in HP’s memristor model. The potential future of memory resistors with respect to artificial intelligence and robotics is briefly discussed.