Optical microscopy suffers from a fundamental resolution limitation arising from the diffractive nature of light. While current solutions to sub-diffraction optical microscopy involve combinations of near-field, non-linear and fine scanning operations, we hereby propose and demonstrate the optical super-microscope (OSM)-a superoscillation-based linear imaging system with far-field working and observation distances-which can image an object in real-time and with sub-diffraction resolution. With our proof-of-principle prototype we report a point spread function with a spot size clearly reduced from the diffraction limit, and demonstrate corresponding improvements in two-point resolution experiments. Harnessing a new understanding of superoscillations, based on antenna array theory, our OSM achieves far-field, sub-diffraction optical imaging of an object without the need for fine scanning, data post-processing or object pre-treatment. Hence the OSM can be used in a wide variety of imaging applications beyond the diffraction limit, including real-time imaging of moving objects.