Cache-aided communications have shown potential for substantial improvement in network performance, which goes far beyond that of traditional caching. Traditional caching (i.e., the bringing and storing of data closer to the end users) is only efficient when a significant portion of the popular files can be locally stored. In cache-aided communications, however, information stored at one user is useful for interference mitigation even if it is requested only by another user. The core idea in cache-aided communication is to use this interferencecancellation opportunity to simultaneously serve multiple users by sending a sum of multiple packets. By creating opportunities for multicasting, the improved performance scales with the accumulated cache size at all users. This is a great advantage for modern networks, where the number of users is typically large, and a small amount of memory can easily be allocated at each user. This article presents the novel techniques of cache-aided communications while focusing on the signal processing aspects that lie in the heart of these schemes. In particular, we examine the three well-studied signal processing problems at the core of cache-aided communications: Resource allocation, beamforming design, and interference mitigation.