This talk deals with the stability analysis of discrete shock profiles for systems of conservation laws. These profiles correspond to approximations of shocks of systems of conservation laws by conservative finite difference schemes. Discontinuous solutions appear naturally in the study of systems of conservation laws, which can model many physical situations, such as gas dynamics. Existence and stability of discrete shock profiles for each stable shock of the approximated system of conservation laws is seen as an improved consistency condition and implies that the finite difference scheme should be able to approach discontinuities fairly precisely. The aim of the talk will be to review some stability results regarding discrete shock profiles and to present a recent effort to extend them. More precisely, most results known up until recently are focused on the stability of discrete shock profiles associated with shocks of small amplitude. The talk will focus on a nonlinear orbital stability result for discrete shock profiles in quite a general setting, where the smallness assumption on the shock's amplitude is replaced by a spectral stability assumption on the linear operator obtained by linearizing the numerical scheme about the discrete shock profile. This nonlinear orbital stability result relies on a precise description of the Green's function of the linearization about discrete profiles. Questo seminario è supportato da PRIN 2022 PNRR 'Heterogeneity on the road: modeling, analysis and control' (Hero-Mac), P2022XJ9SX, CUP B53D23027920001, PNRR Italia Domani, finanziato da UE, programma NextGenerationEU. Link YouTube https://www.youtube.com/watch?v=ZLxIgYVfCQU