Abstract: With the increasing penetration of renewable energy and the proliferation of power-electronics-based equipment, new power systems are increasingly characterized by a high proportion of renewable generation and a high penetration of power electronic devices. These changes are fundamentally reshaping the stability mechanisms, supporting resources, and operational characteristics of power systems. This paper analyzes the evolving features of power-electronics-based source-grid-load-storage systems and the transition of stability support from synchronous-generator-dominated systems to a coordinated framework involving synchronous machines, power electronic devices, and diversified resources. The evolution of stability mechanisms from traditional electromechanical-dominated dynamics to a new stability paradigm characterized by multi-timescale interactions, deep coupling between electromechanical dynamics and control systems, and coordinated support from multiple entities is systematically discussed. Based on the changing requirements of frequency regulation, voltage support, synchronization maintenance, oscillation suppression, and system restoration and reconfiguration, a stability-oriented operational framework for new power systems is proposed. Furthermore, key pathways for enhancing system security and resilience are presented from the perspectives of stability analysis methods, coordinated control architectures, dispatching and operational frameworks, technical standards, market mechanisms, and governance arrangements. The results indicate that adapting to the "dual-high" characteristics requires strengthening the coordinated support of synchronous machines and power electronic resources, promoting source-grid-load-storage interaction, and accelerating the establishment of stability-oriented theoretical, technical, operational, and governance frameworks, thereby providing strong support for energy transition and long-term power system security.