BitTorrent is one of the most important applications in the current Internet. Despite of its interest, we still have little knowledge regarding the connectivity properties of real BitTorrent swarms. In this paper we leverage a dataset including the connectivity information of 250 real torrents and more than 150k peers to carefully study the connectivity properties of peers. The main topology parameters of the studied swarms suggest that they are significantly less resilient than random graphs. The analysis of the peer level connectivity properties reveals that peers continuously change more than half of their neighbours. Furthermore, we also find that a leecher typically keeps stable connections with a handful of neighbours with which it exchanges most of its traffic whereas seeders do not establish long-term connections with any peer so that they can homogeneously distribute chunks among leechers. Finally, we have discovered that a significant portion of the studied peers (45%) have an important locality-biased neighbourhood composition.