Speckle artifacts degrade image quality in virtually all modalities that utilize coherent energy, including optical coherence tomography, reflectance confocal microscopy, ultrasound, and widefield imaging with laser illumination. We present an adversarial deep learning framework for laser speckle reduction, called DeepLSR (https://durr.jhu.edu/DeepLSR), that transforms images from a source domain of coherent illumination to a target domain of speckle-free, incoherent illumination. We apply this method to widefield images of objects and tissues illuminated with a multi-wavelength laser, using light emitting diode-illuminated images as ground truth. In images of gastrointestinal tissues, DeepLSR reduces laser speckle noise by 6.4 dB, compared to a 2.9 dB reduction from optimized non-local means processing, a 3.0 dB reduction from BM3D, and a 3.7 dB reduction from an optical speckle reducer utilizing an oscillating diffuser. Further, DeepLSR can be combined with optical speckle reduction to reduce speckle noise by 9.4 dB. This dramatic reduction in speckle noise may enable the use of coherent light sources in applications that require small illumination sources and high-quality imaging, including medical endoscopy.