The optic disc (OD) and fovea are two important anatomical landmarks of the retina. Localization of OD and fovea plays an important role in retinal image analysis. This paper proposes a novel, efficient and robust method for OD detection and fovea localization. The proposed method consists of two steps. First, we use region proposal network to generate multiple OD region proposals, and then OD’s bounding box is identified as the proposal with the highest probability. Second, we create a region of interest containing the fovea based on the geometrical relationship between OD and fovea, and then employ a three-level cascaded convolutional neural network to locate the fovea, which is treated as a regression problem. The proposed method is trained and evaluated on three sets of fundus images (the first dataset contains 1955 fundus images and the other two are publicly available datasets respectively containing 516 and 1200 fundus images). According to 5-fold cross-validation experiments, our OD detection accuracy is respective 99.59%, 100%, and 100% for the three datasets and the corresponding fovea localization accuracy is respective 97.8%, 99.03%, and 99.25%. The proposed method is found to be robust and effective in OD detection and fovea localization, and its accuracy is comparable, or even superior to representative state-of-the-art algorithms.