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Although temporarily distracted by a little $280 million acquisition, Last.fm has gone ahead today and launched Last.fm Music, the company's first Facebook app.
You don't need a Last.fm account to use it: it'll create a personalized, streaming radio station for all comers and find your musical compatibility with other users. I'm playing around with it now and it's great. The only drawback is that Facebook's ranking system seems to favor the most popular apps: being only a few days behind rival iLike may cost them.
See also: 30+ Awesome Apps for Facebook