Wireless communication is rapidly evolving, and future wireless communications (6G and beyond) will be more heterogeneous, multi-layered, and complex, which poses challenges to traditional communications. Adaptive technologies in traditional communication systems respond to environmental changes by modifying system parameters and structures on their own and are not flexible and agile enough to satisfy requirements in future communications. To tackle these challenges, we propose a novel self-evolving communication framework, which consists of three layers: data layer, information layer, and knowledge layer. The first two layers allow communication systems to sense environments, fuse data, and generate a knowledge base for the knowledge layer. When dealing with a variety of application scenarios and environments, the generated knowledge is subsequently fed back to the first two layers for communication in practical application scenarios to obtain self-evolving ability and enhance the robustness of the system. In this paper, we first highlight the limitations of current adaptive communication systems and the need for intelligence, automation, and self-evolution in future wireless communications. We overview the development of self-evolving technologies and conceive the concept of self-evolving communications with its hypothetical architecture. To demonstrate the power of self-evolving modules, we compare the performances of a communication system with and without evolution. We then provide some potential techniques that enable self-evolving communications and challenges in implementing them.