Several peer-to-peer systems for live streaming have been recently deployed (e.g. CoolStreaming, PPLive, SopCast). These all rely on distributed, epidemic-style dissemination mechanisms. Despite their popularity, the fundamental performance trade-offs of such mechanisms are still poorly understood. In this paper we propose several results that contribute to the understanding of such trade-offs. Specifically, we prove that the so-called random peer, latest useful chunk mechanism can achieve dissemination at an optimal rate and within an optimal delay, up to an additive constant term. This qualitative result suggests that epidemic live streaming algorithms can achieve near-unbeatable rates and delays. Using mean-field approximations, we also derive recursive formulas for the diffusion function of two schemes referred to as latest blind chunk, random peer and latest blind chunk, random useful peer. Finally, we provide simulation results that validate the above theoretical results and allow us to compare the performance of various practically interesting diffusion schemes terms of delay, rate, and control overhead. In particular, we identify several peer/chunk selection algorithms that achieve near-optimal performance trade-offs. Moreover, we show that the control overhead needed to implement these algorithms may be reduced by restricting the neighborhood of each peer without substantial performance degradation.