The weakness of magnetic effects at optical frequencies is directly related to the lack of symmetry between electric and magnetic charges. Natural materials cease to exhibit appreciable magnetic phenomena at rather low frequencies and become unemployable for practical applications in optics. For this reason, historically important efforts were spent in the development of artificial materials. The first evidence in this direction was provided by split-ring resonators in the microwave range. However, the efficient scaling of these devices towards the optical frequencies has been prevented by the strong ohmic losses suffered by circulating currents. With all of these considerations, artificial optical magnetism has become an active topic of research, and particular attention has been devoted to tailor plasmonic metamolecules generating magnetic hot spots. Several routes have been proposed in these directions, leading, for example, to plasmon hybridization in 3D complex structures or Fano-like magnetic resonances. Concurrently, with the aim of electromagnetic manipulation at the nanoscale and in order to overcome the critical issue of heat dissipation, alternative strategies have been introduced and investigated. All-dielectric nanoparticles made of high-index semiconducting materials have been proposed, as they can support both magnetic and electric Mie resonances. Aside from their important role in fundamental physics, magnetic resonances also provide a new degree of freedom for nanostructured systems, which can trigger unconventional nanophotonic processes, such as nonlinear effects or electromagnetic field localization for enhanced spectroscopy and optical trapping.