A memristor by definition relates the charge q and the magnetic flux Φ in a circuit and compliments the resistor R, a capacitor C and an inductor L, as an ingredient of electrical circuits The existence of the memristor as the fourth ideal circuit element was predicted in 1971 based on symmetry arguments, but was not clearly experimentally demonstrated until 2008, in a nanoelectronic TiO2 system. The memristor may prove to be a replacement for the transistor in a future nano-chip integrated circuit. The response of the memristor (M), memristor-capacitor (MC) and memristor-inductor (ML) circuits are presented. The memristor has hysteretic current-voltage characteristics which is its hallmark. It is demonstrated theoretically that the ideal MC (ML) circuit undergoes non-exponential charge (current) decay with two time scales. A simple linear drift model is employed to show that these unusual properties are closely related to the memristor’s internal dynamics. A sophisticated MATLAB based circuit simulation tool, the Berkeley Model and Algorithm Prototyping Platform (MAPP), is then used to perform DC, transient and homotopy analyses of memristor circuits, especially the Resistive Random Access Memory (RRAM) device. MAPP has the capability of treating the compact device memristor model properly, as mathematically well posed and can deal with any model, that can be expressed as an ordinary differential.