Nonlinear optical processes in solid-state systems are typically weak but the strength of the nonlinear interactions can be significantly increased in nanophotonic structures providing a local enhancement of the optical fields. This thesis deals with three experiments in this thematic area. The Kerr nonlinearity of 2D Ruddlesden-Popper-phase (2D RPP) lead halide perovskite flakes is investigated by means of the Z-scan method, after which the flakes are combined with an array of Al nanoantennae to form a nonlinear metasurface. Both nonlinear absorption and
refraction are strongly enhanced near the exciton ground state of the 2D RPP with peak
values of beta_eff = -256 cm/MW and n_2 = +- 10^-13 m^2/W comparable to top values in the
literature. The combined metasurface leads to a further enhancement of the Kerr nonlinearity
attributed to the strong near-fields of the nanoantennae, however, with a complicated
saturation behaviour analysed separately. Moreover, a partial hybridisation of the exciton
and plasmon modes is found. Wavelength-sized cones etched into a GaN layer are studied for their improvement of frequency
mixing efficiencies in the visible-NIR range compared to an unstructured GaN layer. As a preliminary result, the second-harmonic generation (SHG) efficiency is found to be increased by a factor of 10. In future samples with smaller cones, pumped at the magnetic
dipole resonance, a larger efficiency increase is expected. Frequency mixing in a metasurface of Au nanoantennae is investigated with the aim of working towards sources of visible-NIR photon pairs with tailored spectral properties. Frequency scans to map out the joint spectral density are conducted, which will allow for testing the spectral properties of improved future antenna designs. The expected spontaneous parametric down-conversion (SPDC) count rates in the GaN and Au nanoantenna samples are estimated via the stimulated emission tomography (SET) technique.