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Memristors are assembled in crossbar arrays with data bits encoded by the resistance of individual cells. Despite the proposed high density and excellent scalability, the sneak\u2010path current causing cross interference impedes their practical applications. Therefore, developing novel architectures to mitigate sneak\u2010path current and improve efficiency, reliability, and stability may benefit next\u2010generation storage\u2010class memory (SCM). Moreover, conventional digital computers face the von\u2010Neumann bottleneck and the slowdown of transistors\u2019 scaling, imposing a big challenge to hardware artificial intelligence. Memristive crossbar features colocation of memory and processing units, as well as superior scalability, making it a promising candidate for hardware accelerating machine learning and neuromorphic computing. Herein, first, crossbar architecture is introduced. Then, for storage, the origin of sneak\u2010path current is reviewed and techniques to mitigate this issue from the angle of materials and circuits are discussed. Computing wise, the applications of memristive crossbars in both machine learning and neuromorphic computing are surveyed, focusing on the structure of unit cells, the network topology, and the learning types. 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