Threshold-switching memristors (TSMs) are emerging as key enablers for hardware spiking neural networks, offering intrinsic spiking dynamics, sub-pJ energy consumption, and nanoscale footprints ideal for brain-inspired computing at the edge. This review provides a comprehensive examination of how TSMs emulate diverse spiking behaviors-including oscillatory, leaky integrate-and-fire (LIF), Hodgkin-Huxley (H-H), and stochastic dynamics-and how these features enable compact, energy-efficient neuromorphic systems. We analyze the physical switching mechanisms of redox and Mott-type TSMs, discuss their voltage-dependent dynamics, and assess their suitability for spike generation. We review memristor-based neuron circuits regarding architectures, materials, and key performance metrics. At the system level, we summarize bio-inspired neuromorphic platforms integrating TSM neurons with visual, tactile, thermal, and olfactory sensors, achieving real-time edge computation with high accuracy and low power. Finally, we critically examine key challenges-such as stochastic switching origins, device variability, and endurance limits-and propose future directions toward reconfigurable, robust, and scalable memristive neuromorphic architectures.