In this paper, we show that many well-known chaotic maps can be generated by discretizing the equations of memristor or nonlinear resistor circuits using the Euler method or the central difference method.These examples show that the dynamics of a wide variety of nonlinear maps, such as those found in engineering, physics, chemistry, biology, and ecological systems, are closely related to the discretized memristor or nonlinear resistor circuit equations. Furthermore, the discretized memristor circuit equations also propose the new modified or simplified version of the well-known chaotic maps. We also propose the generalized extended memristor with non-volatility property. To satisfy the non-volatility property, the $v-i$ characteristic of the generalized extended memristor is defined by two bounded functions, namely the resistive-fuse function and the saturation function. Using this element, the discretized two-element memristor circuits can generate any two-dimensional chaotic map. The computer simulations in this paper show that the discretization of the memristor or nonlinear resistor circuit equations is one of the most promising methods to find interesting chaotic maps. Furthermore, some of the discretized three-dimensional circuit equations clearly show the topological structure of the chaotic attractors.