This talk presents a class of coverage optimization problems that find application in optimal deployment of robotic networks and distributed spatial estimation. Robotic sensors can service events that occur in a spatial domain, improve the efficiency of data collection, adapt to changes in the environment, and provide a robust response to individual failures. We formalize the coverage task using tools from geometric optimization and computational geometry. A careful analysis of the resulting utility functions leads us to propose various gradient descent algorithms for distributed optimization. The resulting closed-loop behavior is adaptive, distributed, verifiably correct, and amenable to asynchronous implementation. Our analysis of these problems is based on concepts and methods from systems and control, distributed and geometric optimization, and facility location.