State-of-the-art methods in high-resolution three-dimensional optical microscopy require that the focus be scanned through the entire region of interest. However, an analysis of the physics of the light-sample interaction reveals that the Fourier-space coverage is independent of depth. Here we show that, by solving the inverse scattering problem for interference microscopy, computed reconstruction yields volumes with a resolution in all planes that is equivalent to the resolution achieved only at the focal plane for conventional high-resolution microscopy. In short, the entire illuminated volume has spatially invariant resolution, thus eliminating the compromise between resolution and depth of field. We describe and demonstrate a novel computational image-formation technique called interferometric synthetic aperture microscopy (ISAM). ISAM has the potential to broadly impact real-time three-dimensional microscopy and analysis in the fields of cell and tumour biology, as well as in clinical diagnosis where in vivo imaging is preferable to biopsy.