  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 this paper we
review the approaches that have been followed so far to model the enrichment of
the ICM in a cosmological context. While our presentation will be focused on
the role played by hydrodynamical simulations, we will also discuss other
approaches based on semi-analytical models of galaxy formation, also critically
discussing pros and cons of the different methods. We will first review the
concept of the model of chemical evolution to be implemented in any
chemo-dynamical description. We will emphasise how the predictions of this
model critically depend on the choice of the stellar initial mass function, on
the stellar life-times and on the stellar yields. We will then overview the
comparisons presented so far between X-ray observations of the ICM enrichment
and model predictions. We 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. We will conclude by highlighting the open questions in this
study and the direction of improvements for cosmological chemo-dynamical models
of the next generation.
