Caching is an efficient way to reduce network traffic congestion during peak hours, by storing some content at the user's local cache memory, even without knowledge of user's later demands. Maddah-Ali and Niesen proposed a two-phase (placement phase and delivery phase) coded caching strategy for broadcast channels with cache-aided users. This paper investigates the same model under the constraint that content is placed uncoded within the caches, that is, when bits of the files are simply copied within the caches. When the cache contents are uncoded and the users' demands are revealed, the caching problem can be connected to an index coding problem. This paper focuses on deriving fundamental performance limits for the caching problem by using tools for the index coding problem that were either known or are newly developed in this work. First, a converse bound for the caching problem under the constraint of uncoded cache placement is proposed based on the "acyclic index coding converse bound". This converse bound is proved to be achievable by the Maddah-Ali and Niesen's scheme when the number of files is not less than the number of users, and by a newly derived index coding achievable scheme otherwise. The proposed index coding achievable scheme is based on distributed source coding and strictly improves on the widely used "composite (index) coding" achievable bound and its improvements, and is of independent interest. An important consequence of the findings of this paper is that advancements on the coded caching problem posed by Maddah-Ali and Niesen are thus only possible by considering strategies with coded placement phase. A recent work by Yu et al. has however shown that coded cache placement can at most half the network load compared to the results presented in this paper.