Over the last few years, Cloud Radio Access Network (C-RAN) has arisen as a transformative architecture for 5G cellular networks that brings the flexibility and agility of cloud computing to wireless communications. At the same time, content caching in wireless networks has become an essential solution to lower the content-access latency and backhaul traffic loading, which translate into user Quality of Experience (QoE) improvement and network cost reduction. In this article, a novel Cooperative Hierarchical Caching (CHC) framework in C-RAN is introduced where contents are jointly cached at the BaseBand Unit (BBU) and at the Radio Remote Heads (RRHs). Unlike in traditional approaches, the cache at the BBU, cloud cache, presents a new layer in the cache hierarchy, bridging the latency/capacity gap between the traditional edge-based and core-based caching schemes. Trace-driven simulations reveal that CHC yields up to 80% improvement in cache hit ratio, 21% decrease in average content-access latency, and 20% reduction in backhaul traffic load compared to the edge-only caching scheme with the same total cache capacity. Before closing the article, several challenges and promising opportunities for deploying content caching in C-RAN are highlighted towards a content-centric mobile wireless network.