In this paper, we provide a comprehensive definition and classification of various sources of uncertainty within the fields of structural dynamics, system identification, and structural health monitoring (SHM), with a primary focus on the latter. Utilizing the classical input-output system representation as a main contextual framework, we present a taxonomy of uncertainties, intended for consistent classification of uncertainties in SHM applications: (i) input uncertainty; (ii) model form uncertainty; (iii) model parameter/variable uncertainty; (iv) measurement uncertainty; and (v) inherent variability. We then critically review methods and algorithms that address these uncertainties in the context of key SHM tasks: system identification and model inference, model updating, accounting for environmental and operational variability (EOV), virtual sensing, damage identification, and prognostic health management. A benchmark shear frame model with hysteretic links is employed as a running example to illustrate the application of selected methods and algorithmic tools. Finally, we discuss open challenges and future research directions in uncertainty quantification for SHM.