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Magnetic tunneling junction (MTJ)-based programmable logic devices have been proposed and studied for future reconfigurable and nonvolatile computation devices and systems. Spin transfer torque (STT)-based switching has advantages in device scaling compared to the field-switching mechanism. However, the previously proposed MTJ logic devices have operated independently and, therefore, are limited to only basic logic operations. Consequently, the MTJ device has only been used as an ancillary device, rather than the main computation device. As a result, the full benefits of MTJ-based computation have not been explored. New designs are needed to accelerate the development of the MTJ-based logic devices. Specifically the realization of direct communication between the MTJ devices is crucial to fully utilize the MTJ devices in the circuits to implement more advanced logic functions. In this paper, new MTJ-based spintronic logic units (building blocks) for spintronic circuits using the STT switching mechanism have been proposed and investigated, which includes the designs of a basic STT-MTJ logic cell, a direct communication between the MTJ logic cells, a three-MTJ logic unit and a spintronic logic circuit acting as an arithmetic logic unit.