We will overview the models commonly used for the simulation of matter at the microscopic scale. These are model of quantum mechanics, or, when simplified, of classical mechanics, that are applied to the description of molecular systems and crystalline solids.
We will give a flavour of the mathematical questions raised by such models, of the numerical challenges, and the way both are currently addressed. A special emphasis will be laid on the coupling of these microscopic models with the continuum description of matter, and how this coupling is implemented for multiscale numerical simulations.