Funding: UK Engineering and Physical Sciences Research Council through grant EP/P030017/1.Deep learning has revolutionised microscopy, enabling automated means for image classification, tracking and transformation. Going beyond machine vision, deep learning has recently emerged as a universal and powerful tool to address challenging and previously untractable image recovery problems. In seeking accurate, learned means of inversion, these advances have transformed conventional deep learning methods to those cognisant of the underlying physics of image formation, enabling robust, efficient and accurate recovery even in severely ill-posed conditions. In this Perspective, we explore the emergence of physics-informed deep learning that will enab...