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As the fabrication technology node shrinks down to 90nm or below, high standby power becomes one of the major critical issues for CMOS high-speed computing circuits (e.g. logic and cache memory) due to the high leakage currents. A number of non-volatile storage technologies such as FeRAM, MRAM, PCRAM and RRAM and so on, are under investigation to bring the non-volatility into the logic circuits and then eliminate completely the standby power issue. Thanks to its infinite endurance, high switching/sensing speed and easy 3D integration after CMOS process, MRAM is considered as the most promising one. Numerous logic circuits based on MRAM technology have been proposed and prototyped in the last years. In this paper, we present an overview and current status of these logic circuits and discuss their potential applications in the future from both the physics and architecture points of view.