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Federated learning enables cooperative training among massively distributed clients by sharing their learned local model parameters. However, with increasing model size, deploying federated learning requires a large communication bandwidth, which limits its deployment in wireless networks. To address this bottleneck, we introduce a residual-based federated learning framework (ResFed), where residuals rather than model parameters are transmitted in communication networks for training. In particular, we integrate two pairs of shared predictors for the model prediction in both server-to-client and client-to-server communication. By employing a common prediction rule, both locally and globally updated models are always fully recoverable in clients and the server. We highlight that the residuals only indicate the quasi-update of a model in a single inter-round, and hence contain more dense information and have a lower entropy than the model, comparing to model weights and gradients. Based on this property, we further conduct lossy compression of the residuals by sparsification and quantization and encode them for efficient communication. The experimental evaluation shows that our ResFed needs remarkably less communication costs and achieves better accuracy by leveraging less sensitive residuals, compared to standard federated learning. For instance, to train a 4.08 MB CNN model on CIFAR-10 with 10 clients under non-independent and identically distributed (Non-IID) setting, our approach achieves a compression ratio over 700X in each communication round with minimum impact on the accuracy. To reach an accuracy of 70%, it saves around 99% of the total communication volume from 587.61 Mb to 6.79 Mb in up-streaming and to 4.61 Mb in down-streaming on average for all clients.