In the modern era of digital transformation recent advancements have given rise to the concept of twinning, that is, the development of digital representations of physical assets and engineering models in a parallel virtual space. The essence of these virtual representations is the establishment of a continual two-way interaction between the physical and digital counterparts. Digital twins can be employed to simulate hypothetical or extreme scenarios for the system of interest, proving beneficial in an array of tasks including monitoring, control, preventive maintenance, and decision support for engineers and stakeholders. This thesis refers to the process of delivering digital representations that mirror and possibly interact with real-world assets as virtualization and delves into the virtualization of physical assets that can be represented as parameterized dynamical systems. This encompasses deriving a set of digital models that can reproduce the behavior of their physical counterparts using available monitoring information. The methods developed in this work focus on the aspect of model order reduction for fusing computational models with monitoring data that are acquired from the physical system. Such a fusion can only be achieved on the basis of affordable computational models, rendering the aspect of reduction of paramount significance. The two-way interconnection between the physical and virtual systems leads to the following perplexity: on the one hand, computational models form critical elements for introducing engineering knowledge into the resulting digital representations. On the other hand, the requirement for ever-increasing precision implies a higher demand for resources and time-intensive computations, thus compromising efficiency when (near) real-time decisions rely on evaluations of the said virtual models, especially in Structural Health Monitoring (SHM) applications. This work addresses this impasse by focusing on deriving low-dimensional models that can be used for rapid computation while sufficiently reproducing the dynamic behavior of the high-fidelity physical system. The latter is achieved by imprinting the underlying physics into the obtained representation. The derived low-dimensional surrogate, termed a Reduced-Order Model (ROM), strikes a balance between accurate response estimates and efficient model evaluations and acts as an enabler of virtualization for SHM, digital twinning, and decision-making.