Abstract. This talk is concerned with several spectral problems for differential operators arising from topological photonic crystals, which were inspired by the success of topological insulators in condensed matter physics and have been developed to transport optical wave energy in a stable manner. I will present mathematical theory for the existence of edge modes in joint topological photonic crystals formed by two periodic media with different topological natures. Special focus will be on the periodic layered media (1D), the waveguide (1.5D), and the honeycomb structures (2D), where different techniques are developed to investigate the edge modes mathematically. I will also discuss the computation of the spectral problems for these operators, especially on the maximizing the spectral band gap and the inverse design of Dirac point for topological photonic crystals.