Networks portray a multitude of interactions through which people meet, ideas are spread and infectious diseases propagate within a society. Identifying the most efficient `spreaders' in a network is an important step towards optimizing the use of available resources and ensuring the more efficient spread of information. Here we show that, in contrast to common belief, there are plausible circumstances where the best spreaders do not correspond to the most highly connected or the most central people. Instead, we find that the most efficient spreaders are those located within the core of the network as identified by the k-shell decomposition analysis, and that when multiple spreaders are considered simultaneously the distance between them becomes the crucial parameter that determines the extent of the spreading. Furthermore, we show that infections persist in the high-k shells of the network in the case where recovered individuals do not develop immunity. Our analysis should provide a route for an optimal design of efficient dissemination strategies.