Engineered materials exhibit amazing and useful out-of-equilibrium properties. Some are soft but tough; others can harvest waste heat to produce electricity. Their properties often depend on how the materials are processed; during processing they can exhibit complex flow behavior unlike that of simple fluids. Classical descriptions like the Navier–Stokes equation or Hookean elasticity do not capture the mechanics of such materials. Instead, modeling emergent complex behavior requires simultaneous dynamical descriptions on both macroscopic and microscopic length scales. Such multiscale modeling relies on physical insight; the examples discussed here, which use minimal mathematics, show that the growing field is ripe for contributions from physicists, mathematicians, materials scientists, and engineers.