Memristors are a general name for a set of emerging resistive switching technologies. These two terminal devices are characterized by a varying resistance, which is controlled by the voltage or current applied to them. The resistance state of a memristor is nonvolatile, and as such makes memristors attractive candidates for use as novel memory elements. Apart from their use for memory applications, the use of memristors in logic circuits is widely researched. A class of logic circuits named ‘stateful logic’, where the logic state of the inputs and outputs is stored in the form of resistance, is a promising approach for carrying out logic computations within memory. This chapter discusses the use of non-polar memristors, a type of memristors whose resistance depends only on the magnitude of the voltage across its terminals, for performing stateful logic operations. A design methodology is presented to allow structured development of stateful logic gates, and backed by a demonstration of the design process of OR and XOR gates using non-polar memristors.