Silicon carbide (SiC) is emerging rapidly in novel photonic applications thanks to its unique photonic properties facilitated by the advances of nanotechnologies such as nanofabrication and nanofilm transfer. This review paper will start with the introduction of exceptional optical properties of silicon carbide. Then, a key structure, i.e., silicon carbide on insulator stack (SiCOI), is discussed which lays solid fundament for tight light confinement and strong light-SiC interaction in high quality factor and low volume optical cavities. As examples, microring resonator, microdisk and photonic crystal cavities are summarized in terms of quality (Q) factor, volume and polytypes. A main challenge for SiC photonic application is complementary metal-oxide-semiconductor (CMOS) compatibility and low-loss material growth. The state-of-the-art SiC with different polytypes and growth methods are reviewed and a roadmap for the loss reduction is predicted for photonic applications. Combining the fact that SiC possesses many different color centers with the SiCOI platform, SiC is also deemed to be a very competitive platform for future quantum photonic integrated circuit applications. Its perspectives and potential impacts are included at the end of this review paper.