This study provides a comprehensive review of domain adaptation (DA) techniques in vibration-based structural health monitoring (SHM). As data-driven models increasingly support the assessment of civil structures, the persistent challenge of transferring knowledge across varying geometries, materials, and environmental conditions remains a major obstacle. DA offers a systematic approach to mitigate these discrepancies by aligning feature distributions between simulated, laboratory, and field domains while preserving the sensitivity of damage-related information. Drawing on more than sixty representative studies, this paper analyzes the evolution of DA methods for SHM, including statistical alignment, adversarial and subdomain learning, physics-informed adaptation, and generative modeling for simulation-to-real transfer. The review summarizes their contributions and limitations across bridge and building applications, revealing that while DA has improved generalization significantly, key challenges persist: managing domain discrepancy, addressing data scarcity, enhancing model interpretability, and enabling adaptability to multiple sources and time-varying conditions. Future research directions emphasize integrating physical constraints into learning objectives, developing physics-consistent generative frameworks to enhance data realism, establishing interpretable and certifiable DA systems for engineering practice, and advancing multi-source and lifelong adaptation for scalable monitoring. Overall, this review consolidates the methodological foundation of DA for SHM, identifies existing barriers to generalization and trust, and outlines the technological trajectory toward transparent, physics-aware, and adaptive monitoring systems that support the long-term resilience of civil infrastructure.