Recently, it has been demonstrated that memristors can be utilized as logic gates, control switches as well as memory elements. In this paper, we analyze the different AND, OR, and NOT logic gates which are based on memristors. In addition, a novel design for a memristor-based switch is presented, which can be used in the peripheral read/write circuits of the memristor-based memory. Moreover, methods of consecutive read with long refresh intervals and fast write for the proposed design are also discussed. Another highlight of this work is the analysis of the proposed memristor- based crossbar architecture which has a series of excellent features, such as good-compatibility, high-density, non-volatility, low-power, and good-scalability. Simulation results also show that the proposed memory array has superior performances compared to other memristor-based arrays proposed in the existing technical literature.