Coded caching and delivery is studied taking into account the correlations among the contents in the library. Correlations are modeled as common parts shared by multiple contents; that is, each file in the database is composed of a group of subfiles, where each subfile is shared by a different subset of files. The number of files that include a certain subfile is defined as the level of commonness of this subfile. First, a correlation-aware uncoded caching scheme is proposed, and it is shown that the optimal placement for this scheme gives priority to the subfiles with the highest levels of commonness. Then a correlation- aware coded caching scheme is presented, and the cache capacity allocated to subfiles with different levels of commonness is optimized in order to minimize the delivery rate. The proposed correlation-aware coded caching scheme is shown to remarkably outperform state-of-the-art correlation-ignorant solutions, indicating the benefits of exploiting content correlations in coded caching and delivery in networks.