6G wireless networks are foreseen to speed up the convergence of the physical and cyber worlds and to enable a paradigm-shift in the way we deploy and exploit communication networks. Machine learning, in particular deep learning (DL), is going 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 in tackling various challenges in wireless network design. Specifically, we propose transformer-based solutions for massive multiple-input multiple-output (MIMO) systems and various semantic communication problems in 6G networks. Finally, we discuss key challenges and open issues in transformer-based solutions, and identify future research directions for their deployment in intelligent 6G networks.