Over the past decade, the bulk of wireless traffic has shifted from speech to content. This shift creates the opportunity to cache part of the content in memories closer to the end users, for example in base stations. Most of the prior literature focuses on the reduction of load in the backhaul and core networks due to caching, i.e., on the benefits caching offers for the wireline communication link between the origin server and the caches. In this paper, we are instead interested in the benefits caching can offer for the wireless communication link between the caches and the end users. To quantify the gains of caching for this wireless link, we consider an interference channel in which each transmitter is equipped with an isolated cache memory. Communication takes place in two phases, a content placement phase followed by a content delivery phase. The objective is to design both the placement and the delivery phases to maximize the rate in the delivery phase in response to any possible user demands. Focusing on the three-user case, we show that through careful joint design of these phases, we can reap three distinct benefits from caching: a load balancing gain, an interference cancellation gain, and an interference alignment gain. In our proposed scheme, load balancing is achieved through a specific file splitting and placement, producing a particular pattern of content overlap at the caches. This overlap allows to implement interference cancellation. Further, it allows us to create several virtual transmitters, each transmitting a part of the requested content, which increases interference-alignment possibilities.