Panergy Software's docXConverter v3.2 ($19.95) can convert Appleworks 5 and 6 Word Processing documents to RTF (though it has been reported that it can only handle documents which contain only text, not those which include images or frames). The latest version of the free LibreOffice has announced that it can open AppleWorks 6 Word Processing documents and an ability to open ClarisWorks documents has been reported: it does appear to be able to handle at least some embedded images.
Draw Documents can be opened directly only in EazyDraw (the more expensive version from their website, not the cheaper one from the Mac App Store, and you will need v4.0 'Retro' not the most recent version) or Intaglio (though this appears not to be able to handle very complex drawings). LibreOffice will open simple Draw documents but I found it very unreliable with complex ones.
Nothing will open Database or Paint documents (though the beta version of the latest LibreOffice will open Paint documents - sort of...).
If you have upgraded to Lion/Mountain Lion/Mavericks and have AppleWorks documents you cannot open, then you should be able to install Snow Leopard on an external hard disk and start from that (provided that the Mac did not come with anything higher than Snow Leopard originally installed); and success has been reported in running Snow Leopard Server under emulation in Parallels - the method is described here: Snow Leopard Server is available in the Apple Store (by telephone) at a reduced price (at the time of writing) and is preferable to using ordinary Snow Leopard which is quite tricky. I haven't tried this myself: use at your own discretion. You can then run AppleWorks as an emergency measure.
You may find this article on 'Abandoning AppleWorks' of interest:
http://rfwilmut.net/aw