In this talk, I review recent efforts on the development of registration methods for parametric model order reduction (MOR), with emphasis on advection-dominated flows. In computer vision and pattern recognition, registration refers to the process of finding a parametric transformation that aligns two datasets; in model order reduction, registration methods seek a parametric bijection that tracks coherent structures (e.g., shocks, shear layers) of the solution field. We integrate registration in the offline/online model reduction framework to tackle problems with parameter-dependent discontinuities. Our approach combines registration with three additional building blocks: (i) an hyper-reduced least-squares Petrov-Galerkin (LSPG) reduced-order model, to estimate the mapped solution; (ii) a parametric mesh adaptation procedure to build a parsimonious yet accurate representation of the solution field; and (iii) a multi-fidelity strategy to reduce offline training costs. We present numerical results for several two-dimensional inviscid compressible flows, to show the potential of the method.