The objective of this thesis is to significantly reduce the computational cost associated with numerical simulations governed by partial differential equations. For this purpose, we consider reduced-order models (ROMs), which typically consist of a training stage, in which high-fidelity solutions are collected to define a low-dimensional trial subspace, and a prediction stage, where this data-driven trial subspace is then exploited to achieve fast or real-time simulations. The first contribution of this thesis concerns the modeling of gas flows in both hydrodynamic and rarefied regimes. In this work, we develop a new reduced-order approximation of the Boltzmann-BGK equation, based on Proper Orthogonal Decomposition (POD) in the training stage and on the Galerkin method in the prediction stage. We investigate the simulation of unsteady flows containing shock waves, boundary layers and vortices in 1D and 2D. The results demonstrate the stability, accuracy and significant computational speedup factor delivered by the ROM with respect to the high-fidelity model. The second topic of this thesis deals with the optimal transport problem and its applications to model order reduction. In particular, we propose to use the optimal transport theory in order to analyze and enrich the training database containing the high-fidelity solution snapshots. Reproduction and prediction of unsteady flows, governed by the 1D Boltzmann-BGK equation, show the improvement of the accuracy and reliability of the ROM resulting from these two applications. Finally, the last contribution of this thesis concerns the development of a domain decomposition method based on the Discontinuous Galerkin method. In this approach, the ROM approximates the solution where a significant dimensionality reduction can be achieved while the high-fidelity model is employed elsewhere. The Discontinuous Galerkin method for the ROM offers a simple way to recover the global solution by linking local solutions through numerical fluxes at cell interfaces. The proposed method is evaluated for parametric problems governed by the quasi-1D and 2D Euler equations. The results demonstrate the accuracy of the proposed method and the significant reduction of the computational cost with respect to the high-fidelity model.