We show that, in contrast to popular belief, sub-wavelength information can be recovered from the far-field of an optical image, thereby overcoming the loss of information embedded in decaying evanescent waves. The only requirement is that the image is known to be sparse, a specific but very general and wide-spread property of signals which occur almost everywhere in nature. The reconstruction method relies on newly-developed compressed sensing techniques, which we adapt to optical super-resolution and sub-wavelength imaging. Our approach exhibits robustness to noise and imperfections. We provide an experimental proof-of-principle by demonstrating image recovery at a spatial resolution 5-times higher than the finest resolution defined by a spatial filter. The technique is general, and can be extended beyond optical microscopy, for example, to atomic force microscopes, scanning-tunneling microscopes, and other imaging systems.