This lecture focuses on fundamentals in crystal and parametric optics. It aims at giving guidelines and tools for understanding the main concepts as well as the design, characterization and use of crystals for optical parametric generation. The following main aspects are detailed. •Constitutive relations and Maxwell equations. •Classification of the nonlinear interactions using the corpuscular approach: fusion and splitting involving three or four photons, spontaneous and stimulated processes. •Calculation of the electric susceptibility from the Lorentz model: perturbation approach leading to the definition of the different orders of the electric susceptibility, wavelength dispersion, intrinsic symmetries (Kleinman and ABDP), implications of spatial symmetry on the susceptibility tensors (Neumann principle). •Tensors algebra and calculation of the first, second and third order polarizations. •Modelling of the macroscopic nonlinearities of matter from the microscopic scale using the bond charge model and ab initio calculations, Miller index. •Basics in linear crystal optics: propagation equation, index surface, birefringence, double refraction, eigenmodes. •Amplitude equations in the nonlinear regime, Manley-Rowe relations. •Calculation of the effective coefficient based on the field tensor formalism. •Types and topology of collinear and non-collinear Birefringence Phase-Matching and Quasi-Phase-Matching in bulk media and whispering-gallery-mode resonators. •Integration of the amplitude equations and calculation of the conversion efficiencies associated with second harmonic generation (SHG), direct and cascaded third harmonic generation (THG), spontaneous parametric down-conversion (SPDC), optical parametric amplification (OPA), optical parametric generation (OPG), optical parametric oscillation (OPO). •Angular, spectral and thermal acceptances. •Spatial and temporal walk-off effects. •Techniques of characterization of nonlinear crystals: refractive indices, phase-matching and quasi-phase-matching loci, conversion efficiencies, magnitudes and relative signs of the nonlinear coefficients, acceptances. •The main nonlinear crystals for optical parametric generation, from ultraviolet to THz.