Microscopy is the workhorse of the physical and life sciences, producing crisp images of everything from atoms to cells well beyond the capabilities of the human eye. However, the analysis of these images is frequently little better than automated manual marking. Here, we revolutionize the analysis of microscopy images, extracting all the information theoretically contained in a complex microscope image. Using a generic, methodological approach, we extract the information by fitting experimental images with a detailed optical model of the microscope, a method we call Parameter Extraction from Reconstructing Images (PERI). As a proof of principle, we demonstrate this approach with a confocal image of colloidal spheres, improving measurements of particle positions and radii by 100x over current methods and attaining the maximum possible accuracy. With this unprecedented resolution, we measure nanometer-scale colloidal interactions in dense suspensions solely with light microscopy, a previously impossible feat. Our approach is generic and applicable to imaging methods from brightfield to electron microscopy, where we expect accuracies of 1 nm and 0.1 pm, respectively. ; Comment: 34 pages, 19 figures (7p 3f main text, 27p 16f appended supplemental)