Mobile phones have facilitated the creation of field-portable, cost-effective imaging and sensing technologies that approach laboratory-grade instrument performance. However, the optical imaging interfaces of mobile phones are not designed for microscopy and produce distortions in imaging microscopic specimens. Here, we report on the use of deep learning to correct such distortions introduced by mobile-phone-based microscopes, facilitating the production of high-resolution, denoised, and color-corrected images, matching the performance of benchtop microscopes with high-end objective lenses, also extending their limited depth of field. After training a convolutional neural network, we successfully imaged various samples, including human tissue sections and Papanicolaou and blood smears, where the recorded images were highly compressed to ease storage and transmission. This method is applicable to other low-cost, aberrated imaging systems and could offer alternatives for costly and bulky microscopes, while also providing a framework for standardization of optical images for clinical and biomedical applications.