Recent high-resolution observations of debris disks by ALMA and direct imaging have revealed a rich variety of structures, such as gaps, spirals arms, and warps. In analogy with the studies of the asteroid and Kuiper belts, investigating the structure of debris disks can provide unique insight into the architecture and evolution of planetary systems. Most existing dynamical processes that have already been put forward as a means of sculpting debris disk morphologies usually treat the disk as a collection of massless particles, subject only to the gravity of point-mass perturbers such as planets or stellar companions. However, as I will discuss in this talk, self-gravity of debris disks can be important for producing some of the observed disk structures. Specifically, I will present results showing that the secular interaction between a single planet and an external, self-gravitating debris disk can lead to the formation of a gap in the disk. This mechanism is quite robust when the disk is less massive than the planet. Applications of these results for explaining observations will be discussed at length, focusing on the HD 107146 system. I will also discuss the implications of our findings for inferring the presence of unseen planets based on observed gaps in debris disks.