The development of medical imaging techniques has greatly supported clinical decision making. However, poor imaging quality, such as non-uniform illumination or imbalanced intensity, brings challenges for automated screening, analysis and diagnosis of diseases. Previously, bi-directional GANs (e.g., CycleGAN), have been proposed to improve the quality of input images without the requirement of paired images. However, these methods focus on global appearance, without imposing constraints on structure or illumination, which are essential features for medical image interpretation. In this paper, we propose a novel and versatile bi-directional GAN, named Structure and illumination constrained GAN (StillGAN), for medical image quality enhancement. Our StillGAN treats low- and high-quality images as two distinct domains, and introduces local structure and illumination constraints for learning both overall characteristics and local details. Extensive experiments on three medical image datasets (e.g., corneal confocal microscopy, retinal color fundus and endoscopy images) demonstrate that our method performs better than both conventional methods and other deep learning-based methods. In addition, we have investigated the impact of the proposed method on different medical image analysis and clinical tasks such as nerve segmentation, tortuosity grading, fovea localization and disease classification.