Device-to-device (D2D) communication is a promising approach to optimize the utilization of air interface resources in 5G networks, since it allows decentralized opportunistic short-range communication. For D2D to be useful, mobile nodes must possess content that other mobiles want. Thus, intelligent caching techniques are essential for D2D. In this paper we use results from stochastic geometry to derive the probability of successful content delivery in the presence of interference and noise. We employ a general transmission strategy where multiple files are cached at the users and different files can be transmitted simultaneously throughout the network. We then formulate an optimization problem, and find the caching distribution that maximizes the density of successful receptions (DSR) under a simple transmission strategy where a single file is transmitted at a time throughout the network. We model file requests by a Zipf distribution with exponent $γ_r$, which results in an optimal caching distribution that is also a Zipf distribution with exponent $γ_c$, which is related to $γ_r$ through a simple expression involving the path loss exponent. We solve the optimal content placement problem for more general demand profiles under Rayleigh, Ricean and Nakagami small-scale fading distributions. Our results suggest that it is required to flatten the request distribution to optimize the caching performance. We also develop strategies to optimize content caching for the more general case with multiple files, and bound the DSR for that scenario.