Voltage-source converters (VSCs) are becoming the dominant interface in modern power systems as renewable generation, energy storage, high-voltage DC transmission, medium-voltage DC networks, electric-vehicle charging, and large flexible loads continue to expand. In VSC-dominated hybrid AC/DC power systems, stability and resilience are increasingly determined by converter controls rather than by the inherent electromechanical behavior of synchronous machines. This dissertation develops modeling, stability-analysis, and resilience-oriented control methods for VSC-based hybrid AC/DC power systems. The first part of the dissertation investigates small-signal AC/DC dynamic coupling across different AC- and DC-terminal control roles, including AC grid-forming/DC grid-following, AC grid-following/DC grid-forming, and dual-port grid-forming operation. The analysis shows how DC-link dynamics, modulation-reference choices, DC-network dynamics, and dual-port control parameters affect oscillatory modes and stability margins. Modular single-input single-output modeling frameworks and physical damping interpretations are developed to support scalable and design-oriented stability analysis. The second part studies large-signal resilience of P-f droop-based AC grid-forming/DC grid-following VSCs under faults, unbalanced black start, load pickup, motor startup, and protection-device interactions. Current-limiting methods, generalized three-phase droop control, distributed load restoration, and converter-aware protection behavior are examined to understand how current-limited VSCs can support practical grid functions during severe disturbances. Together, the results provide control-aware modeling tools, stability insights, and resilience-oriented design guidelines for future hybrid AC/DC grids with high penetrations of VSC-interfaced resources.