It is well known that memristors can be classified into the four classes according to the complexity of their mathematical representation. Furthermore, the four classes of memristors are used to qualitatively simulate many of the experimentally measured pinched hysteresis loops. In this paper, we define the 2-terminal devices, which do not belong to the above four classes of memristors, but have the same non-volatile property as the ideal memristor. We then study the non-volatile mechanism of these devices and the memristors that can retain the previous value of the state even when the driving signal is set to zero. We show that the ideal generic memristors and the generalized 2-terminal devices can have interesting applications: non-volatile multi-valued memories and two-element chaotic oscillators, if we remove the condition that no state change occurs after the zero driving signal. We also show that the 2-terminal devices and the four classes of memristors can exhibit a wide variety of pinched hysteresis loops similar to those measured experimentally. Furthermore, we show that a wide variety of Lissajous curves are possible, depending on whether the direction of the Lissajous curve is clockwise or counterclockwise and which quadrants the Lissajous curve passes through.