6G networks will be required to support higher data rates, improved energy efficiency, lower latency, and more diverse users compared with 5G systems. To meet these requirements, electrically extremely large-scale antenna arrays are envisioned to be key physical-layer technologies. As a consequence, it is expected that some portion of future 6G wireless communications may take place in the radiating near-field (Fresnel) region, in addition to the far-field operation as in current wireless technologies. In this article, we discuss the opportunities and challenges that arise in radiating near-field communications. We begin by discussing the key physical characteristics of near-field communications, where the standard plane-wave propagation assumption no longer holds, and clarifying its implication on the modelling of wireless channels. Then, we elaborate on the ability to leverage spherical wavefronts via beam focusing, highlighting its advantages for 6G systems. We point out several appealing application scenarios which, with proper design, can benefit from near-field operation, including interference mitigation in multi-user communications, accurate localization and focused sensing, as well as wireless power transfer with minimal energy pollution. We conclude by discussing some of the design challenges and research directions that are yet to be explored to fully harness the potential of near-field operation.