Unlike Kazaa, the eDonkey and BitTorrent networks are truly decentralized with no company or single server running the show behind the scenes. That there's no legal entity behind the networks will make shutting them down especially difficult for copyright holders, who will have to target individual users individually rather than the networks as a whole.
Of course, that's not to say that these new networks don't have some very legitimate uses. Already, BitTorrent has been instrumental in aiding in the distribution of Linux ISOs, which are perfectly legal. The network also seems like a perfect distribution medium something like a Doom 3 demo/test, which would undoubtedly bring download servers to their knees.