This paper surveys the emerging paradigm of cloud mobile media. We start with two alternative perspectives for cloud mobile media networks: an end-to-end view and a layered view. Summaries of existing research in this area are organized according to the layered service framework: i) cloud resource management and control in infrastructure-as-a-service (IaaS), ii) cloud-based media services in platform-as-a-service (PaaS), and iii) novel cloud-based systems and applications in software-as-a-service (SaaS). We further substantiate our proposed design principles for cloud-based mobile media using a concrete case study: a cloud-centric media platform (CCMP) developed at Nanyang Technological University. Finally, this paper concludes with an outlook of open research problems for realizing the vision of cloud-based mobile media.