This summer school will provide a thorough introduction to the material standing at the core of the USM programme. Lectures will cover state-of-the-art techniques and methodologies for stochastic modeling, uncertainty propagation and inverse problems, as well as emerging methods for adaptivity and probabilistic learning. A broad array of applications will be presented, ranging from UQ-informed materials design to the stochastic modeling of complex, multiscale physical systems. The workshop is suited in particular for PhD students, postdocs and academic researchers from computational engineering, materials science, mathematics, and statistics.