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The performance of deep networks for medical image analysis is often constrained by limited medical data, which is privacy-sensitive. Federated learning (FL) alleviates the constraint by allowing different institutions to collaboratively train a federated model without sharing data. However, the federated model is often suboptimal with respect to the characteristics of each client's local data. Instead of training a single global model, we propose Customized FL (CusFL), for which each client iteratively trains a client-specific/private model based on a federated global model aggregated from all private models trained in the immediate previous iteration. Two overarching strategies employed by CusFL lead to its superior performance: 1) the federated model is mainly for feature alignment and thus only consists of feature extraction layers; 2) the federated feature extractor is used to guide the training of each private model. In that way, CusFL allows each client to selectively learn useful knowledge from the federated model to improve its personalized model. We evaluated CusFL on multi-source medical image datasets for the identification of clinically significant prostate cancer and the classification of skin lesions.