In naturally occurring swarms – living as well as non-living – local proximity encounters among individuals or particles in the collective facilitate a broad range of emergent phenomena. In the context of robot swarms operating with limited sensing and communication capabilities, this thesis demonstrates how the systematic analysis of inter-robot encounters can enable the swarm to perform useful functions without the presence of a central coordinator. We combine ideas from stochastic geometry, statistical mechanics, and biology to develop mathematical models which characterize the nature and frequency of inter-robot encounters occurring in a robot swarm. These models allow the swarm to perform functions like localization, task allocation, and density regulation, while only requiring individual robots to measure the presence of other robots in the immediate vicinity – either via contact sensors or binary proximity detectors. Moreover, the resulting encounter-based algorithms require no communication among the robots or the presence of a central coordinator, and are robust to individual robot failures occurring in the swarm. Throughout the thesis, experiments conducted on real robot swarms vindicate the idea that inter-robot encounters can be advantageously leveraged by individuals in the swarm. Advisors/Committee Members: Egerstedt, Magnus (advisor), Hutchinson, Seth (committee member), Goldman, Daniel (committee member), Wardi, Yorai (committee member), Shell, Dylan (committee member).