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The characteristic pinched hysteresis behavior of memristors has been reported by stacks of a variety of materials. This paper aims to examine the principles of logic design using such two terminal memristive systems for high performance digital circuit applications. As against logic design with standard CMOS, the benefits of logic design with memristors have been stated. The realization and operation of memristor based AND and OR hybrid logic gates obtained by integrating memristors with standard CMOS logic have been discussed. The IMPLY and MAGIC logic families have been demonstrated by covering MAGIC NOR and NAND logic gate implementation with MAGIC NOR in detail. A qualitative comparison has been drawn towards the end of the paper to conclude on the suitability and application space for each of the logic families studied in this paper. This work also describes the hybrid CMOS-memristive logic family known as MRL (Memristor Ratioed Logic). With the addition of CMOS inverters, this logic family's OR and AND logic gates, which are based on memristive components, are given a full logic structure and signal restoration. The MRL family, in contrast to earlier memristor-based logic families, is compatible with conventional CMOS logic.