It is anticipated that 6G wireless networks will accelerate the convergence of the physical and cyber worlds and enable a paradigm-shift in the way we deploy and exploit communication networks. Machine learning, in particular deep learning (DL), is expected to be one of the key technological enablers of 6G by offering a new paradigm for the design and optimization of networks with a high level of intelligence. In this article, we introduce an emerging DL architecture, known as the transformer, and discuss its potential impact on 6G network design. We first discuss the differences between the transformer and classical DL architectures, and emphasize the transformer's self-attention mechanism and strong representation capabilities, which make it particularly appealing for tackling various challenges in wireless network design. Specifically, we propose transformer-based solutions for various massive multiple-input multiple-output (MIMO) and semantic communication problems, and show their superiority compared to other architectures. Finally, we discuss key challenges and open issues in transformer-based solutions, and identify future research directions for their deployment in intelligent 6G networks.