Abstract: Topological photonics and phononics are rapidly developing fields that apply the principles of topological insulators from condensed matter physics into periodic optical and acoustic media. A central challenge in these systems is to understand how in-gap interface modes emerge at the boundary between two bulk media in distinct topological phases. Such modes are topologically protected and remain robust under a broad class of perturbations. In this talk, we review recent progress in elucidating the mechanisms that give rise to in-gap interface modes across various topological photonic and phononic structures.