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Since the emergence of memristors (or memristive devices), how to integrate them into arrays has been widely investigated. After years of research, memristor crossbar arrays have been proposed and realized with potential applications in nonvolatile memory, logic and neuromorphic computing systems. Despite the promising prospects of memristor crossbar arrays, one of the main obstacles for their development is the so-called sneak-path current causing cross-talk interference between adjacent memory cells and thus may result in misinterpretation which greatly influences the operation of memristor crossbar arrays. Solving the sneak-path current issue, the power consumption of the array will immensely decrease, and the reliability and stability will simultaneously increase. In order to suppress the sneak-path current, various solutions have been provided. So far, some reviews have considered some of these solutions and established a sophisticated classification, including 1D1M, 1T1M, 1S1M (D: diode, M: memristor, T: transistor, S: selector), self-selective and self-rectifying memristors. Recently, a mass of studies have been additionally reported. This review thus attempts to provide a survey on these new findings, by highlighting the latest research progress realized for relieving the sneak-path issue. Here, we first present the concept of the sneak-path current issue and solutions proposed to solve it. Consequently, we select some typical and promising devices, and present their structures and properties in detail. Then, the latest research activities focusing on single-device structures are introduced taking into account the mechanisms underlying these devices. Finally, we summarize the properties and perspectives of these solutions.