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The performance of light microscopes is limited by the stochastic nature of light, which exists in discrete packets of energy known as photons. Randomness in the times that photons are detected introduces shot noise, which fundamentally constrains sensitivity, resolution and speed 1 . Although the long-established solution to this problem is to increase the intensity of the illumination light, this is not always possible when investigating living systems, because bright lasers can severely disturb biological processes 2-4 . Theory predicts that biological imaging may be improved without increasing light intensity by using quantum photon correlations 1,5 . Here we experimentally show that quantum correlations allow a signal-to-noise ratio beyond the photodamage limit of conventional microscopy. Our microscope is a coherent Raman microscope that offers subwavelength resolution and incorporates bright quantum correlated illumination. The correlations allow imaging of molecular bonds within a cell with a 35 per cent improved signal-to-noise ratio compared with conventional microscopy, corresponding to a 14 per cent improvement in concentration sensitivity. This enables the observation of biological structures that would not otherwise be resolved. Coherent Raman microscopes allow highly selective biomolecular fingerprinting in unlabelled specimens 6,7 , but photodamage is a major roadblock for many applications 8,9 . By showing that the photodamage limit can be overcome, our work will enable order-of-magnitude improvements in the signal-to-noise ratio and the imaging speed.