Memristors with controllable resistive switching (RS) behavior have been considered as promising candidates for synaptic devices in next-generation neuromorphic computing. In this work, two-terminal memristors with controllable digital and analog RS behavior are fabricated based on two-dimensional (2D) WSe₂ nanosheets. Under a relatively high operating voltage of 4 V, the memristor demonstrates stable and reliable non-volatile bipolar digital RS with a high switching ratio of 6.3 × 10⁴. On the other hand, under a relatively low operation voltage, the memristor exhibits analog RS with a series of tunable resistance states. The fabricated memristors can work as an artificial synapse with fundamental synaptic functions, such as long-term potentiation (LTP) and depression (LTD) as well as paired-pulse facilitation (PPF). More importantly, the memristor demonstrates high conductance modulation linearity with the calculated nonlinear parameter for conductance as −0.82 in the LTP process, which is beneficial to improving the accuracy of neuromorphic computing. Furthermore, the neuromorphic computing of file types and image recognition can be emulated based on a constructed three-layer artificial neural network (ANN) with a recognition accuracy that can reach up to 95.9% for small digits. In addition, memristors can be used to emulate the learning-forgetting experience of the human brain. Consequently, the memristor based on 2D WSe₂ nanosheets not only exhibits controllable RS behavior but also simulates synaptic functions and is expected to be a potential candidate for future neuromorphic computing applications.