Neuromorphic computing has emerged as an alternative computing paradigm to address the increasing computing needs for data-intensive applications. In this context, resistive random access memory (RRAM) devices have garnered immense interest among the neuromorphic research community due to their capability to emulate intricate neuronal behaviors. RRAM devices excel in terms of their compact size, fast switching capabilities, high ON/OFF ratio, and low energy consumption, among other advantages. This review focuses on the multifaceted aspects of RRAM devices and their application to brain-inspired computing. The review begins with a brief overview of the essential biological concepts that inspire the development of bio-mimetic computing architectures. It then discusses the various types of resistive switching behaviors observed in RRAM devices and the detailed physical mechanisms underlying their operation. Next, a comprehensive discussion on the diverse material choices adapted in recent literature has been carried out, with special emphasis on the benchmark results from recent research literature. Further, the review provides a holistic analysis of the emerging trends in neuromorphic applications, highlighting the state-of-the-art results utilizing RRAM devices. Commercial chip-level applications are given special emphasis in identifying some of the salient research results. Finally, the current challenges and future outlook of RRAM-based devices for neuromorphic research have been summarized. Thus, this review provides valuable understanding along with critical insights and up-to-date information on the latest findings from the field of resistive switching devices towards brain-inspired computing.