In recent years, deep learning has been shown to be one of the leading machine learning techniques for a wide variety of inference tasks. In addition to its mainstream applications, such as classification, it has created transformative opportunities for image reconstruction and enhancement in optical microscopy. Some of these emerging applications of deep learning range from image transformations between microscopic imaging systems to adding new capabilities to existing imaging techniques, as well as solving various inverse problems based on microscopy image data. Deep learning is helping us move toward data-driven instrument designs that blend microscopy and computing to achieve what neither can do alone. This article provides an overview of some of the recent work using deep neural networks to advance computational microscopy and sensing systems, also covering their current and future biomedical applications.