  We review the latest advances in the generation of quantum light with
nonlinear nanoresonators in metasurfaces, which act both as sources of quantum
states and nanoantennas shaping the emitted photons. We outline a general
quantum theory of spontaneous photon-pair generation in arbitrary nonlinear
photonic structures, including nanoresonators and metasurfaces, which provides
an explicit analytical solution for the photon state expressed through the
classical Green function. We formulate the correspondence between the quantum
photon-pair generation and classical sum-frequency process in nonlinear media,
and discuss its application in various contexts, including waveguide circuits
and nanostructures. We also discuss the first experimental results
demonstrating photon-pair generation in a single nonlinear nanoantenna.
