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Caching at the wireless edge is a promising way to boost spectral efficiency and reduce energy consumption of wireless systems. These improvements are rooted in the fact that popular contents are reused, asynchronously, by many users. In this article we first introduce methods to predict the popularity distributions and user preferences, and the impact of erroneous information. We then discuss the two aspects of caching systems, content placement and delivery. We expound the key differences between wired and wireless caching, and outline the differences in the system arising from where the caching takes place (e.g., at base stations or on the wireless devices themselves). Special attention is paid to the essential limitations in wireless caching, and possible trade-offs between spectral efficiency, energy efficiency, and cache size.