Caching is an efficient way to reduce network traffic congestion during peak hours by storing some content at the user's local cache memory without knowledge of later demands. For the shared-link caching model, Maddah-Ali and Niesen (MAN) proposed a two-phase (placement and delivery) coded caching strategy, which is order optimal within a constant factor. However, in the MAN coded caching scheme, each user can obtain the information about the demands of other users, i.e., the MAN coded caching scheme is inherently prone to tampering and spying the activity/demands of other users. In this paper, we formulate an information-theoretic shared-link caching model with private demands, where there are K cache-aided users (which can cache up to M files) connected to a central server with access to N files. Each user requests L files. Our objective is to design a two-phase private caching scheme with minimum load while preserving the information-theoretic privacy of the demands of each user with respect to other users. A trivial solution is the uncoded caching scheme which lets each user recover all the N files, referred to as baseline scheme. For this problem we propose two novel schemes which achieve the information-theoretic privacy of the users' demands while also achieving a non-trivial caching gain over the baseline scheme. The general underlying idea is to satisfy the users' requests by generating a set of coded multicast messages that is symmetric with respect to the library files, such that for each user k, the mutual information between these messages and the demands of all other users given the cache content and the demand of user k is zero. In the first scheme, referred to as virtual-user scheme, we introduce a number of virtual users such that each L-subset of files is demanded by K real or virtual (effective) users and use the MAN delivery to generate multicast messages. From the viewpoint of each user, the set of multicast messages is symmetric over all files even if each single multicast message is not. This scheme incurs in an extremely large sub-packetization. Then, we propose a second scheme, referred to as MDS-based scheme, based on a novel MDS-coded cache placement. In this case, we generate multicast messages where each multicast message contains one MDS-coded symbol from each file in the library and thus is again symmetric over all the files from the viewpoint of each user. The sub-packetization level of the MDS-based scheme is exponentially smaller than that needed by the virtual-user scheme. Compared with the existing shared-link coded caching converse bounds without privacy, the virtual-user scheme is proved to be order optimal with a constant factor when N ≤ LK, or when N ≥ LK and M ≥ N/K. In addition, when M ≥ N/2, both of the virtual-user scheme and the MDS-based scheme are order optimal within a factor of 2.