Weakly supervised semantic segmentation (WSSS) using only image-level labels can greatly reduce the annotation cost and therefore has attracted considerable research interest. However, its performance is still inferior to the fully supervised counterparts. To mitigate the performance gap, we propose a saliency guided self-attention network (SGAN) to address the WSSS problem. The introduced self-attention mechanism is able to capture rich and extensive contextual information but may mis-spread attentions to unexpected regions. In order to enable this mechanism to work effectively under weak supervision, we integrate class-agnostic saliency priors into the self-attention mechanism and utilize class-specific attention cues as an additional supervision for SGAN. Our SGAN is able to produce dense and accurate localization cues so that the segmentation performance is boosted. Moreover, by simply replacing the additional supervisions with partially labeled ground-truth, SGAN works effectively for semi-supervised semantic segmentation as well. Experiments on the PASCAL VOC 2012 and COCO datasets show that our approach outperforms all other state-of-the-art methods in both weakly and semi-supervised settings.