The prospect of creating metamaterials with optical properties greatly exceeding the parameter space accessible with natural materials has been inspiring intense research efforts in nanophotonics for more than a decade. Following an era of plasmonic metamaterials, low-loss dielectric nanostructures have recently moved into the focus of metamaterial-related research. This development was mainly triggered by the experimental observation of electric and magnetic multipolar Mie-type resonances in high-refractive-index dielectric nanoparticles. Silicon in particular has emerged as a popular material choice, due to not only its high refractive index and very low absorption losses in the telecom spectral range, but also its paramount technological relevance. This Review overviews recent progress on metamaterial-inspired silicon nanostructures, including Mie-resonant and off-resonant regimes.