Reduced Order Modelling (ROM) has become a central tool for bridging the gap between high-fidelity models and computational efficiency, especially in regimes where repeated evaluations, real-time constraints, or high-dimensional parameter spaces are involved. In parallel, the growing relevance of data-driven methodologies raises fundamental questions on how to represent, propagate, and quantify uncertainty in reduced models in a principled and interpretable manner. In this contribution, we explore a class of uncertainty quantification (UQ) strategies for reduced models based on Optimal Transport (OT) principles. OT provides a geometrically grounded framework to compare, interpolate, and evolve probability measures, making it particularly well-suited for representing uncertainty in data-driven ROMs. The proposed perspective emphasises the transport of distributions rather than pointwise approximations, allowing reduced models to retain information on epistemic variability induced by limited data, model reduction, or incomplete observations. We discuss how OT-based constructions can be integrated with reduced representations to produce uncertainty-aware surrogates that balance accuracy and efficiency, while remaining flexible with respect to the underlying governing equations or data sources. The resulting framework thus naturally introduces modern probabilistic learning approaches, offering a pathway towards reduced models that are not only fast and accurate, but also statistically informative.