Super-resolution microscopy is already showing huge benefits across the biosciences, but all widely-used techniques require the addition of fluorescent probes. We have demonstrated optical-super-resolution imaging in unlabelled living cells, using the phenomenon of super-oscillation. Super-oscillation is originally a mathematical phenomenon, first described in quantum mechanics. It is widely accepted that any function that is band-limited (in frequency) oscillates no faster (in time) than its fastest Fourier component. However, a band-limited super-oscillatory function may oscillate arbitrarily fast in regions of relatively low intensity. In optics, this means that we can create an arbitrarily small hotspot at the focus of a lens using en...