Nonlinear frequency conversion processes, such as second-harmonic generation (SHG) and spontaneous parametric down-conversion (SPDC), are essential in many applications, including the generation of entangled photons. It's desirable to enhance these processes since the nonlinear effects are generally very small and photonic crystal (PhC) cavities are usually implemented for this purpose thanks to the combination of high quality factors and small footprints. Singly resonant PhC cavities, which support one mode at either the input frequency or the output frequency, are extensively investigated by previous studies, and significant enhancement is observed, while doubly resonant PhC cavities, which support simultaneously one mode at the input frequency and another mode at the output frequency, are not because of the difficulty of designing the two resonant modes, even though the enhancement is supposed to be stronger. This thesis studies the first design of 2D doubly resonant PhC cavities, realized by implementing a bound state in the continuum (BIC), and investigates its applications in SHG and SPDC. The cavities were fabricated on a III-nitride-on-silicon platform with standard semiconductor techniques and characterized on custom optical setups. The doubly resonant condition was fulfilled through passive lithographic tuning and active strain tuning. The SHG was observed with a high conversion efficiency under continuous wave excitation, which is in good agreement with theoretical predictions. Besides, far-field emission patterns with a donut shape and radial polarization at the second harmonic frequency were observed, which also agrees with the BIC confinement theory. The investigation on SPDC was however limited by the sensitivity of classical photodetectors. Nevertheless, the observation of high-efficiency SHG demonstrated the concept of doubly resonant PhC cavities and marked a cornerstone achievement for nonlinear frequency conversion. The fabrication and optical measurement techniques developed in this thesis could also be valuable for future studies.