Weakly supervised semantic segmentation is receiving great attention due to its low human annotation cost. In this paper, we aim to tackle bounding box supervised semantic segmentation, i.e., training accurate semantic segmentation models using bounding box annotations as supervision. To this end, we propose affinity attention graph neural network ( AGNN). Following previous practices, we first generate pseudo semantic-aware seeds, which are then formed into semantic graphs based on our newly proposed affinity Convolutional Neural Network (CNN). Then the built graphs are input to our AGNN, in which an affinity attention layer is designed to acquire the short- and long- distance information from soft graph edges to accurately propagate semantic labels from the confident seeds to the unlabeled pixels. However, to guarantee the precision of the seeds, we only adopt a limited number of confident pixel seed labels for AGNN, which may lead to insufficient supervision for training. To alleviate this issue, we further introduce a new loss function and a consistency-checking mechanism to leverage the bounding box constraint, so that more reliable guidance can be included for the model optimization. Experiments show that our approach achieves new state-of-the-art performances on Pascal VOC 2012 datasets (val: 76.5 percent, test: 75.2 percent). More importantly, our approach can be readily applied to bounding box supervised instance segmentation task or other weakly supervised semantic segmentation tasks, with state-of-the-art or comparable performance among almot all weakly supervised tasks on PASCAL VOC or COCO dataset. Our source code will be available at https://github.com/zbf1991/A2GNN.