Graph neural networks (GNNs) are becoming the de facto method to learn on the graph data and have achieved the state-of-the-art on node and graph classification tasks. However, recent works show GNNs are vulnerable to training-time poisoning attacks -- marginally perturbing edges, nodes, and node features of training graphs can largely degrade the GNN performance. Most previous defenses against such attacks are empirical and are soon broken by adaptive / stronger ones. A few provable defenses provide robustness guarantees, but have large gaps when applied in practice: 1) all restrict the attacker’s capability to only one type of perturbation; 2) all are designed for a particular GNN task; and 3) their robustness guarantees are not 100\% accurate. In this work, we bridge all these gaps by developing PGNNCert, the first certified defense of GNNs against poisoning attacks under arbitrary (edge, node, and node feature) perturbations with deterministic (100\% accurate) robustness guarantees. PGNNCert is also applicable to the two most widely-studied node and graph classification tasks. Extensive evaluations on multiple node and graph classification datasets and GNNs demonstrate the effectiveness of PGNNCert to provably defend against arbitrary poisoning perturbations. PGNNCert is also shown to significantly outperform the state-of-the-art certified defenses against edge perturbation or node perturbation during GNN training.