Towards automated retinal screening, this paper makes an endeavor to simultaneously achieve pixel-level retinal lesion segmentation and image-level disease classification. Such a multi-task approach is crucial for accurate and clinically interpretable disease diagnosis. Prior art is insufficient due to three challenges, i.e., lesions lacking objective boundaries, clinical importance of lesions irrelevant to their size, and the lack of one-to-one correspondence between lesion and disease classes. This paper attacks the three challenges in the context of diabetic retinopathy (DR) grading. We propose Lesion-Net, a new variant of fully convolutional networks, with its expansive path redesigned to tackle the first challenge. A dual Dice loss that leverages both semantic segmentation and image classification losses is introduced to resolve the second challenge. Lastly, we build a multi-task network that employs Lesion-Net as a side-attention branch for both DR grading and result interpretation. A set of 12K fundus images is manually segmented by 45 ophthalmologists for 8 DR-related lesions, resulting in 290K manual segments in total. Extensive experiments on this large-scale dataset show that our proposed approach surpasses the prior art for multiple tasks including lesion segmentation, lesion classification and DR grading.