Using a deep neural network, we demonstrate a digital staining technique, which we term PhaseStain, to transform the quantitative phase images (QPI) of label-free tissue sections into images that are equivalent to the brightfield microscopy images of the same samples that are histologically stained. Through pairs of image data (QPI and the corresponding brightfield images, acquired after staining), we train a generative adversarial network and demonstrate the effectiveness of this virtual-staining approach using sections of human skin, kidney, and liver tissue, matching the brightfield microscopy images of the same samples stained with Hematoxylin and Eosin, Jones' stain, and Masson's trichrome stain, respectively. This digital-staining framework may further strengthen various uses of label-free QPI techniques in pathology applications and biomedical research in general, by eliminating the need for histological staining, reducing sample preparation related costs and saving time. Our results provide a powerful example of some of the unique opportunities created by data-driven image transformations enabled by deep learning.