Next-generation mobile networks promise to support high throughput, massive connectivity, and improved energy efficiency. To achieve these ambitious goals, extremely large-scale antenna arrays (ELAAs) and terahertz communications constitute a pair of promising technologies. This will result in future wireless communications occurring in the near-field regions. To accurately portray the channel characteristics of near-field wireless propagation, spherical wavefront-based models are required and present both opportunities as well as challenges. Following the basics of near-field communications (NFC), we contrast it to conventional far-field communications. Moreover, we cover the key challenges of NFC, including its channel modeling and estimation, near-field beamfocusing, as well as hardware design. Our numerical results demonstrate the potential of NFC in improving the spatial multiplexing gain and positioning accuracy. Finally, a suite of open issues are identified for motivating future research.