The risks associated with the poor maintenance of aging built infrastructure have fostered the increasingly frequent implementation of long-term Structural Health Monitoring (SHM) systems in recent years. This has necessitated the adaptation of classical damage identification techniques to a new Big Data context, where computational efficiency plays a major role in the timely identification of early-stage structural malfunctioning. In this context, data-driven damage identification techniques have become predominant, even though their efficacy usually limits to damage detection, while information on the localization and severity of the pathology can be inferred only in some specific circumstances. Instead, model-driven damage identification techniques can provide full damage identification capabilities by linking the monitored structural response with the intrinsic mechanical properties of certain structural elements. Nevertheless, such techniques typically involve the inverse calibration of a computationally intensive numerical model, making their use incompatible with continuous SHM schemes. In this light, the use of computationally efficient surrogate models shows great potential for the development of digital twins (DTs) compatible with continuous SHM systems within the Statistical Pattern Recognition paradigm. These systems are capable of receiving continuous streams of monitoring data, conducting condition inference of certain structural elements, and fusing the time series of identified model parameters with data-based damage sensitive features for comprehensive damage identification. In this context, this work presents the latest developments of surrogate model-based DTs for model-driven damage identification of large civil engineering structures, both from an algorithmic and a methodological standpoint. The potential, limitations, and perspectives for future developments of these techniques are showcased through several real-world structures, including both historical constructions and bridges.