Huge infrastructure such as the dam was categorised as critical infrastructure and should be maintained for continuous operation during its intended lifetime. Over the years, instability due to climate change has become a serious threat to dams and therefore requires regular monitoring of structural integrity. This article provides a comprehensive review of dam safety and structural health monitoring under numerous environmental conditions (e.g. normal conditions, flood and earthquake). Interestingly, this study analyses the impact of climate change on the partial failure and collapse of dams based on historical data. A comparison between monitoring approaches, such as the monitoring method, the type of instrumentation and the method of data analysis, was further discussed to understand the complementarity of each approach. Several future directions were outlined to highlight high-risk scenarios and interesting research areas for dam safety and monitoring to support dam sustainability. This will help to develop effective tools and techniques for accurate maintenance based on the issues at hand. A deeper understanding of the relationship between environmental factors (hydraulic impacts) and structural behavioural changes will greatly benefit the various studies on basic fluid-structure interactions, comprehensive emergency plans under the influence of climate change, reliable methods for dam maintenance and retrofitting, and holistic dam safety monitoring plans (surveillance approaches and disaster preparedness). In short, early detection of dam risks, analysis of expected failure and thorough interventions based on regular monitoring are essential for the sustainability and resilience of dam infrastructure.