Azureus and its descendents are based on this code. Versions 1 - 4 of their software were Open Source via various licenses.
The BitTorrent company was formed in 2004. It is also the name of the original, very first BT client, created by Bram Cohen in 2001. 'BitTorrent' (BT) is the name of a file sharing protocol.
If you like bells and whistles, aka Geek Meter in the RED, you want BiglyBT. I personally use the latest version of Transmission along with an up-to-date blocking list. In 2010, Vuze was purchased by Spigot, became proprietary, was parallel named µTorrent, is now considered ADWARE and should be avoided. NO, this is not the torrenting client you're looking for.
