We consider the problem of spectrum sharing in device-to-device communication systems. Inspired by the recent optimality condition for treating interference as noise, we define a new concept of "information-theoretic independent sets" (ITIS), which indicates the sets of links for which simultaneous communication and treating the interference from each other as noise is information-theoretically optimal (to within a constant gap). Based on this concept, we develop a new spectrum sharing mechanism, called "information-theoretic link scheduling" (ITLinQ), which at each time schedules those links that form an ITIS. We first provide a performance guarantee for ITLinQ by characterizing the fraction of the capacity region that it can achieve in a network with sources and destinations located randomly within a fixed area. Furthermore, we demonstrate how ITLinQ can be implemented in a distributed manner, using an initial 2-phase signaling mechanism which provides the required channel state information at all the links. Through numerical analysis, we show that distributed ITLinQ can outperform similar state-of-the-art spectrum sharing mechanisms, such as FlashLinQ, by more than a 100% of sum-rate gain, while keeping the complexity at the same level. Finally, we discuss a variation of the distributed ITLinQ scheme which can also guarantee fairness among the links in the network and numerically evaluate its performance.