By merging cloud computing into the RAN, C-RAN has been foreseen as a prospective 5G wireless systems architecture. Due to the innovative move of migrating the baseband processing functionalities to the centralized cloud baseband unit pool, C-RAN is anticipated to reduce energy consumption significantly to be a green RAN. Moreover, with the cloud-based architecture, lots of new functionalities and RAN designs are ready to be incorporated, which redefines the RAN as a flexible RAN. In this article, we review the recent advances of exploiting cloud computing to form a green and flexible C-RAN from two cloud-based properties: centralized processing and the software-defined environment. For the centralized processing property, we include coordinated multipoint and limited fronthaul capacity, multicasting, and CSI issues in C-RAN. For the software-defined environment property, we summarize elastic service scaling, functionality splitting, and functionality extension. We also include some of our recent research results and discuss several open challenges.