We will discuss some recent developments of the theory of a contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows. In a first result, in collaboration with Geng Chen and Sam Krupa, we provide some extensions of the Bressan theory for uniqueness of BV solutions in 1D. We show that for 2 × 2 systems, the technical condition, known as bounded variations on space- like curve, is not needed for the uniqueness result. Moreover, we extend the result to a weak/BV stability result (in the spirit of the weak/strong principle of Dafermos) allowing wild perturbations fulfilling only the so-called strong trace property. In a second work in collaboration with Moon-Jin Kang, we consider the stability of 1D viscous shocks for the compressible Navier-Stokes equation, uniformly with respect to the viscosity (JEMS 21'). Thanks to the uniformity with respect to the viscosity, the result can be extended to the Euler equation (the associated inviscid model).This provides a stability result which holds in the class of wild perturbations of inviscid limits of solutions to Navier-Stokes, without any regularity restriction, not even the strong trace property (Inventiones 21'). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem. Finally, we will present a first multi-D result obtained with Moon-Jin Kang and Yi Wang. We show the stability of contact discontinuities without shear, in the class of inviscid limits of compressible Fourier-Navier-Stokes equation. Note that it is still unknown whether non-uniqueness results can be obtained via convex integration for this special kind of singularity.