Fluorescent molecules move and rotate as they transition between states and emit photons that we can detect. How much information can we recover about the position, orientation, and motion of fluorescent molecules by measuring these photons? Can we design imaging systems that recover as much information as possible by making optimal measurements? In this dissertation, we develop mathematical models of this class of experiments, efficient reconstruction and visualization schemes, and methods for designing instruments that acquire optimal samples. We use these methods to propose and demonstrate the first spatio-angular imaging system that can measure the three-dimensional position and orientation of fluorescent molecules that densely label biological structures. We validate our techniques by successfully reconstructing known spatio-angular objects, and we are beginning to use spatio-angular microscopy to explore new biological structures. Our proof of concept paves the way towards microscopes that can encode, measure, and reconstruct fundamentally new information from fluorescent molecules. We discuss existing applications and speculate on future directions.