Neuromorphic computing systems employing artificial synapses and neurons are expected to overcome the limitations of the present von Neumann computing architecture in terms of efficiency and bandwidth limits. Traditional neuromorphic devices have used 3D bulk materials, and thus, the resulting device size is difficult to be further scaled down for high density integration, which is required for highly integrated parallel computing. The emergence of two-dimensional (2D) materials offers a promising solution, as evidenced by the surge of reported 2D materials functioning as neuromorphic devices for next-generation computing. In this review, we summarize the 2D materials and their heterostructures to be used for neuromorphic computing devices, which could be classified by the working mechanism and device geometry. Then, we survey neuromorphic device arrays and their applications including artificial visual, tactile, and auditory functions. Finally, we discuss the current challenges of 2D materials to achieve practical neuromorphic devices, providing a perspective on the improved device performance, and integration level of the system. This will deepen our understanding of 2D materials and their heterojunctions and provide a guide to design highly performing memristors. At the same time, the challenges encountered in the industry are discussed, which provides a guide for the development direction of memristors.