Two-dimensional (2D) materials present an exciting opportunity for devices and systems beyond the von Neumann computing architecture paradigm due to their diversity of electronic structure, physical properties, and atomically-thin, van der Waals structures that enable ease of integration with conventional electronic materials and silicon-based hardware. All major classes of non-volatile memory (NVM) devices have been demonstrated using 2D materials, including their operation as synaptic devices for applications in neuromorphic computing hardware. Their atomically-thin structure, superior physical properties, i.e., mechanical strength, electrical and thermal conductivity, as well as gate-tunable electronic properties provide performance advantages and novel functionality in NVM devices and systems. However, device performance and variability as compared to incumbent materials and technology remain major concerns for real applications. Ultimately, the progress of 2D materials as a novel class of electronic materials and specifically their application in the area of neuromorphic electronics will depend on their scalable synthesis in thin-film form with desired crystal quality, defect density, and phase purity.