The talk, intended for a general audience, will survey some challenging mathematical and numerical problems in contemporary materials science. Questions such as the passage from the microscale to the macroscale, the insertion of uncertainties, defects and heterogeneities in the models, will be examined. We will discuss the interesting issues raised for mathematical analysis (theory of partial differential equations, stochastic processes, homogenization theory) and for numerical analysis (finite element methods, discrete to continuum, Monte Carlo methods, etc).