Monitoring-data-driven reliability assessment of bridge infrastructure uses structural health monitoring, inspection, load testing, bridge weigh-in-motion, visual/non-destructive testing (NDT) data and digital twins to characterize evolving condition and failure risk. This review asks how monitoring data are transformed into reliability inputs, how model updating methods complement one another, how limited data support time-dependent reliability and rare-event estimation, and how updated reliability supports maintenance decisions. A Scopus-based search of English journal papers published from 2020 through the 30 June 2026 search cutoff was conducted using keywords on bridges, monitoring data, model updating, reliability/risk assessment and maintenance. After thematic screening, 114 papers were coded and critically synthesized, comprising 56 core bridge studies and 58 supplementary studies retained only for method transfer or comparison. The review establishes a technical chain from monitoring data to structural state characterization, model updating, uncertainty quantification, reliability/risk assessment and maintenance decision-making, and formulates a probabilistic framework based on observation models, state/parameter updating, posterior predictive reliability and risk- and loss-based decision-making. The findings show that finite-element updating contributes physical interpretability, Bayesian updating supports evidence fusion and posterior uncertainty, filtering enables online recursion, surrogate modelling and machine learning reduce computational burden, and digital twins are meaningful only when data assimilation, uncertainty propagation, reliability calculation and decision feedback are integrated. Future work should strengthen traceable data-to-limit-state mappings, explicit treatment of model-form and measurement errors, tail failure-probability estimation under sparse monitoring data, and closed-loop reliability-informed bridge maintenance.