Hajek and Zhu (Stoch Syst 1(2):246–273, 2011) recently showed that the BitTorrent protocol can become unstable when peers depart immediately after downloading all pieces of a file. In light of this result, Zhou et al. (2011) propose bundling swarms together, allowing peers to exchange pieces across different swarms, and claim that such “universal swarms” can increase BitTorrent’s stability region. In this work, we formally characterize the stability region of universal swarms and show that they indeed exhibit excellent properties. In particular, bundling allows a single seeder with limited upload capacity to serve an arbitrary number of disjoint swarms if the arrival rate of peers in each swarm is lower than the seeder upload capacity. Our result also shows that the stability region is insensitive to peers’ upload capacity, piece selection policies and number of swarms.