Federated learning is a distributed machine learning paradigm that allows multiple clients to collaboratively train a shared model with their local data. Nonetheless, conventional federated learning algorithms often struggle to generalize well due to the ubiquitous domain shift across clients. In this work, we consider a challenging yet realistic federated learning scenario where the training data of each client originates from different domains. We address the challenges of domain shift by leveraging the technique of prompt learning, and propose a novel method called Federated Dual Prompt Tuning (Fed-DPT). Specifically, Fed-DPT employs a pre-trained vision-language model and then applies both visual and textual prompt tuning to facilitate domain adaptation over decentralized data. Extensive experiments of Fed-DPT demonstrate its significant effectiveness in domain-aware federated learning. With a pre-trained CLIP model (ViT-Base as image encoder), the proposed Fed-DPT attains 68.4% average accuracy over six domains in the DomainNet dataset, which improves the original CLIP by a large margin of 14.8%.