The use of 2D materials to improve the capabilities of electronic devices is a promising strategy that has recently gained much interest in both academia and industry. However, while the research in 2D metallic and semiconducting materials is well established, detailed knowledge and applications of 2D insulators are still scarce. In this paper, the presence of resistive switching (RS) in multilayer hexagonal boron nitride (h-BN) is studied using different electrode materials, and a family of h-BN-based resistive random access memories with tunable capabilities is engineered. The devices show the coexistence of forming free bipolar and threshold-type RS with low operation voltages down to 0.4 V, high current on/off ratio up to 106, and long retention times above 10 h, as well as low variability. The RS is driven by the grain boundaries (GBs) in the polycrystalline h-BN stack, which allow the penetration of metallic ions from adjacent electrodes. This reaction can be boosted by the generation of B vacancies, which are more abundant at the GBs. To the best of our knowledge, h-BN is the first 2D material showing the coexistence of bipolar and threshold RS, which may open the door to additional functionalities and applications.