As of this writing (and much is surely still up in the air), Napster has lost a major legal battle and is seeking to settle the suit by offering to pay $1 billion out of their future profits. In order to pay those fees, Napster is planning to institute a monthly fee for users.
Napster's technology requires the use of their central servers, which undoubtedly made them a convenient legal target. It is unclear what the effect of fully peer to peer systems (systems with no central server) such as Gnutella or Audiogalaxy will be on the availability of free music sharing on the internet. If Gnutella or Audiogalaxy develops into a viable (and free) alternative, but with no central server to take the legal heat, Napster's proposed business model (paid access) might fail.
Another potential competitor or successor to Napster is OpenNAP. Since most existing Napster clients can select what server they connect to, and since there is a free software clone of Napster server, anyone on the Internet can theoretically open up a small competitor or successor to Napster. If this is true, then any future death of Napster might only mean a temporary slowdown in the growth of filesharing.