The edge computing paradigm presents a promising solution for low-latency and high-throughput data processing, which is critical for future life-changing applications, such as autonomous driving and metaverse. Concurrently, the current radio access network (RAN) is evolving toward virtualization, intelligence, and flexibility, making it an ideal edge computing platform. However, the full potential of such a synergistic system has not yet been unleashed due to the lack of coordination between communication and computing. To better support diverse applications, this article proposes a communication-computing co-design approach for the next-generation RAN to embrace computing in a built-in manner. At a high level, we overview the features of communication and computing, identify the efforts and limitations of the current system, and introduce the designed architecture. From the perspective of communication and computing coordination, key technologies are introduced, including internetworking for computing terminals, traffic steering, and mobility management. A use case referring to mobility management is studied to verify the effectiveness of the built-in approach. We outline some critical challenges, thereby indicating future potential research directions in future RAN.