Federated learning provides a communication-efficient and privacy-preserving training process by enabling learning statistical models with massive participants without accessing their local data. Standard federated learning techniques that naively minimize an average loss function are vulnerable to data corruptions from outliers, systematic mislabeling, or even adversaries. In this article, we address this challenge by proposing Auto-weighted Robust Federated Learning (ARFL), a novel approach that jointly learns the global model and the weights of local updates to provide robustness against corrupted data sources. We prove a learning bound on the expected loss with respect to the predictor and the weights of clients, which guides the definition of the objective for robust federated learning. We present an objective that minimizes the weighted sum of empirical risk of clients with a regularization term, where the weights can be allocated by comparing the empirical risk of each client with the average empirical risk of the best ( p ) clients. This method can downweight the clients with significantly higher losses, thereby lowering their contributions to the global model. We show that this approach achieves robustness when the data of corrupted clients is distributed differently from the benign ones. To optimize the objective function, we propose a communication-efficient algorithm based on the blockwise minimization paradigm. We conduct extensive experiments on multiple benchmark datasets, including CIFAR-10, FEMNIST, and Shakespeare, considering different neural network models. The results show that our solution is robust against different scenarios, including label shuffling, label flipping, and noisy features, and outperforms the state-of-the-art methods in most scenarios.