The ever-increasing volume of complex data poses significant challenges to conventional sequential global processing methods, highlighting their inherent limitations. This computational burden has catalyzed interest in neuromorphic computing, particularly within artificial neural networks (ANNs). In pursuit of advancing neuromorphic hardware, researchers are focusing on developing computation strategies and constructing high-density crossbar arrays utilizing history-dependent, multistate nonvolatile memories tailored for multiply-accumulate (MAC) operations. However, the real-time collection and processing of massive, dynamic data sets require an innovative computational paradigm akin to that of the human brain. Spiking neural networks (SNNs), representing the third generation of ANNs, are emerging as a promising solution for real-time spatiotemporal information processing due to their event-based spatiotemporal capabilities. The ideal hardware supporting SNN operations comprises artificial neurons, artificial synapses, and their integrated arrays. Currently, the structural complexity of SNNs and spike-based methodologies requires hardware components with biomimetic behaviors that are distinct from those of conventional memristors used in deep neural networks. These distinctive characteristics required for neuron and synapses devices pose significant challenges. Developing effective building blocks for SNNs, therefore, necessitates leveraging the intrinsic properties of the materials constituting each unit and overcoming the integration barriers. This review focuses on the progress toward memristor-based spiking neural network neuromorphic hardware, emphasizing the role of individual components such as memristor-based neurons, synapses, and array integration along with relevant biological insights. We aim to provide valuable perspectives to researchers working on the next generation of brain-like computing systems based on these foundational elements.