The integration of robotics into Structural Health Monitoring (SHM) is rapidly reshaping how infrastructure is assessed and maintained. This review critically examines the current landscape of robotic technologies applied in SHM, with a specific focus on their implementation within the Saudi Arabian construction sector. It explores recent advancements in robotic platforms, such as unmanned aerial vehicles (UAVs), wall-climbing robots, and AI-driven inspection systems, and assesses their roles in damage detection, vibration monitoring, and real-time diagnostics. In addition to outlining technological capabilities, this paper identifies major adoption challenges related to system readiness, regulatory gaps, workforce limitations, and environmental constraints. Drawing on comparative experiences in the healthcare, energy, and legal domains, this review extracts cross-sectoral insights that offer practical guidance for accelerating robotic integration in SHM. This paper concludes by outlining research gaps and actionable recommendations to support scholars, policymakers, and industry professionals in advancing robotics-based monitoring in complex infrastructure environments.