Recent breakthroughs in brain-inspired computing promise to address a wide range of problems from security to healthcare. However, the current strategy of implementing artificial intelligence algorithms using conventional silicon hardware is leading to unsustainable energy consumption. Neuromorphic hardware based on electronic devices mimicking biological systems is emerging as a low-energy alternative, although further progress requires materials that can mimic biological function while maintaining scalability and speed. As a result of their diverse unique properties, atomically thin two-dimensional (2D) materials are promising building blocks for next-generation electronics including nonvolatile memory, in-memory and neuromorphic computing, and flexible edge-computing systems. Furthermore, 2D materials achieve biorealistic synaptic and neuronal responses that extend beyond conventional logic and memory systems. Here, we provide a comprehensive review of the growth, fabrication, and integration of 2D materials and van der Waals heterojunctions for neuromorphic electronic and optoelectronic devices, circuits, and systems. For each case, the relationship between physical properties and device responses is emphasized followed by a critical comparison of technologies for different applications. We conclude with a forward-looking perspective on the key remaining challenges and opportunities for neuromorphic applications that leverage the fundamental properties of 2D materials and heterojunctions.