Problems with dominant advection, discontinuities, traveling features, or shape variations are widespread in computational mechanics. However, classical linear model reduction and interpolation methods typically fail to reproduce even relatively small parameter variations, making the reduced models inefficient and inaccurate. This talk introduces some new techniques to deal with type of problems. Among them, I will discuss a novel reduced-order modeling approach based on the Radon-Cumulative-Distribution transform (RCDT). We show that this non-linear transformation can significantly improve the dimensionality of proper orthogonal decomposition (POD) reconstructions and is capable of interpolating accurately some advection-dominated phenomena. The method is tested on various test cases in multi-phase fluid dynamics.