From bird flocks to fish schools, animals move together and respond to their environment in remarkable ways; their natural collective motion patterns appear well choreographed and their collective survival strategies seem ingenious. These animal group behaviors inspire design for groups of mobile, sensor-equipped robots, where demanding cooperative sensing tasks, such as exploration and mapping in uncertain, dynamic environments in land, sea, air, or space, find their analogue in natural group behaviors, such as foraging and feeding. However, bio-inspired design of this kind is not immediate because the natural mechanisms are not well understood. Mathematical modeling and analysis play a critical role in addressing this joint challenge to explain the enabling mechanisms in animal groups and to define provable mechanisms for robotic groups. A common framework based on notions of synchrony will be used to discuss connections among spatial pattern, information passing, and collective behavior in robot and animal networks. Applications to be presented include the design of an adaptive ocean observation system using a fleet of underwater robotic vehicles and an investigation of motion and decision-making in bird flocks and fish schools.