Advances in the field of plasmonics, that is, nanophotonics based on optical properties of metal nanostructures, paved the way for the development of ultrasensitive biological sensors and other devices whose operating principles are based on localization of an electromagnetic field at the nanometer scale. However, high dissipative losses of metal nanostructures limit their performance in many modern areas, including metasurfaces, metamaterials, and optical interconnections, which required the development of new devices that combine them with high refractive index dielectric nanoparticles. Resulting metal-dielectric (hybrid) nanostructures demonstrated many superior properties from the point of view of practical application, including moderate dissipative losses, resonant optical magnetic response, strong nonlinear optical properties, which made the development in this field the vanguard of the modern light science. This review is devoted to the current state of theoretical and experimental studies of hybrid metal-dielectric nanoantennas and nanostructures based on them, capable of selective scattering light waves, amplifying and transmitting optical signals in the desired direction, controlling the propagation of such signals, and generating optical harmonics.