Information-theoretic security -- widely accepted as the strictest notion of security -- relies on channel coding techniques that exploit the inherent randomness of the propagation channels to significantly strengthen the security of digital communications systems. Motivated by recent developments in the field, this paper aims at a characterization of the fundamental secrecy limits of wireless networks. Based on a general model in which legitimate nodes and potential eavesdroppers are randomly scattered in space, the intrinsically secure communications graph (iS-graph) is defined from the point of view of information-theoretic security. Conclusive results are provided for the local connectivity of the Poisson iS-graph, in terms of node degrees and isolation probabilities. It is shown how the secure connectivity of the network varies with the wireless propagation effects, the secrecy rate threshold of each link, and the noise powers of legitimate nodes and eavesdroppers. Sectorized transmission and eavesdropper neutralization are explored as viable strategies for improving the secure connectivity. Lastly, the maximum secrecy rate between a node and each of its neighbours is characterized, and the case of colluding eavesdroppers is studied. The results help clarify how the spatial density of eavesdroppers can compromise the intrinsic security of wireless networks.