Recently, there has been increasing interest in the challenge of how to discriminatively vectorize graphs. To address this, we propose a method called Iterative Graph Self-Distillation (IGSD) which learns graph-level representation in an unsupervised manner through instance discrimination using a self-supervised contrastive learning approach. IGSD involves a teacher-student distillation process that uses graph diffusion augmentations and constructs the teacher model using an exponential moving average of the student model. The intuition behind IGSD is to predict the teacher network representation of the graph pairs under different augmented views. As a natural extension, we also apply IGSD to semi-supervised scenarios by jointly regularizing the network with both supervised and self-supervised contrastive loss. Finally, we show that finetuning the IGSD-trained models with self-training can further improve the graph representation power. Empirically, we achieve significant and consistent performance gain on various graph datasets in both unsupervised and semi-supervised settings, which well validates the superiority of IGSD.