Title: Personalized Subgraph Federated Learning. Abstract: In real-world scenarios, subgraphs of a larger global graph may be distributed across multiple devices or institutions, and only locally accessible due to privacy restrictions, although there may be links between them. Recently proposed subgraph Federated Learning (FL) methods deal with those missing links across private local subgraphs while distributively training Graph Neural Networks (GNNs) on them. However, they have overlooked the inevitable heterogeneity among subgraphs, caused by subgraphs comprising different communities of a global graph, therefore, consequently collapsing the incompatible knowledge from local GNN models trained on heterogeneous graph distributions. To overcome such a limitation, we introduce a new subgraph FL problem, personalized subgraph FL, which focuses on the joint improvement of the interrelated local GNN models rather than learning a single global GNN model, and propose a novel framework, FEDerated Personalized sUBgraph learning (FED-PUB), to tackle it. A crucial challenge in personalized subgraph FL is that the server does not know which subgraph each client has. FED-PUB thus utilizes functional embeddings of the local GNNs using random graphs as inputs to compute similarities between them, and use them to perform weighted averaging for server-side aggregation. Further, it learns a personalized sparse mask at each client to select and update only the subgraph-relevant subset of the aggregated parameters. We validate FED-PUB for its subgraph FL performance on six datasets, considering both non-overlapping and overlapping subgraphs, on which ours largely outperforms relevant baselines.