The study of the metal enrichment of the intra-cluster and inter-galactic media (ICM and IGM) represents a direct means to reconstruct the past history of star formation, the role of feedback processes and the gas-dynamical processes which determine the evolution of the cosmic baryons. In these lectures, I review the approaches that have been followed so far to model the enrichment of the ICM in a cosmological context. While the presentation will be focused on the role played by hydrodynamical simulations, I will also discuss other approaches based on semi-analytical models of galaxy formation, also critically discussing pros and cons of the different methods. I will first review the concept of the model of chemical evolution to be implemented in any chemo-dynamical description. I will emphasise how the predictions of this model critically depends on the choice of the stellar initial mass function, on the stellar lifetimes and on the stellar yields. I will then overview the comparisons presented so far between X-ray observations of the ICM enrichment and model predictions. I will show how the most recent chemo-dynamical models are able to capture the basic features of the observed metal content of the ICM and its evolution. I will conclude by highlighting the open questions in this study and the direction of improvements for cosmological chemo-dynamical models of the next generation.
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