With many fantastic properties, memristive devices have been proposed as top candidate for next-generation memory and neuromorphic computing chips. Significant research progresses have been made in improving performance of individual memristive devices and in demonstrating functional applications based on small-scale memristive crossbar arrays. However, practical deployment of large-scale traditional metal oxides based memristive crossbar array has been challenging due to several issues, such as high-power consumption, poor device reliability, low integration density and so on. To solve these issues, new materials that possess superior properties are required. Two-dimensional (2D) layered materials exhibit many unique physical properties and show great promise in solving these challenges, further providing new opportunities to implement practical applications in neuromorphic computing. Here, recent research progress on 2D layered materials based memristive device applications is reviewed. We provide an overview of the progresses and challenges on how 2D layered materials are used to solve the issues of conventional memristive devices and to realize more complex functionalities in neuromorphic computing. Besides, we also provide an outlook on exploitation of unique properties of 2D layered materials and van der Waals heterostructures for developing new types of memristive devices and artificial neural mircrocircuits.