In cache-aided networks, due to the multicast nature of wireless signals, coded multicast delivery results in significant load reductions that are proportional to the aggregate cache size. State of the art caching schemes exploit exact content reuse across the network, which, due to the personalized nature of the content consumed by users, can lead to limited efficiency of the scheme. In this work, the cache-aided coded multicast setting is extended to specifically exploit the correlation among the content files. The load over the shared link is notably reduced by jointly compressing the content during the caching and delivery phases. Two classes of correlation-aware achievable schemes are proposed, both of which are shown to significantly outperform existing correlation-unaware solutions, while approaching the limiting rate-memory trade-off.