Cellular networks are one of the corner stones of our information-driven society. However, existing cellular systems have been seriously challenged by the explosion of mobile data traffic, the emergence of machine-type communications and the flourish of mobile Internet services. In this article, we propose CONCERT (CONvergence of Cloud and cEllulaR sysTems), a converged edge infrastructure for future cellular communications and mobile computing services. The proposed architecture is constructed based on the concept of control/data (C/D) plane decoupling. The data plane includes heterogeneous physical resources such as radio interfacing equipment, computational resources, and software-defined switches. The control plane jointly coordinates physical resources to present them as virtual resources, over which software-defined services including communications, computing, and management can be deployed in a flexible manner. Moreover, we introduce new designs for physical resources placement and task scheduling, so that CONCERT can overcome the drawbacks of the existing baseband-up centralization approach and better facilitate innovations in next-generation cellular networks. These advantages are demonstrated with application examples on the radio access networks (RANs) with C/D decoupled air interface, delay sensitive machine-type communications, and real-time mobile cloud gaming. We also discuss some fundamental research issues arising with the proposed architecture to illuminate future research directions.