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Given an arbitrary triangle mesh, is it possible to automatically generate a UV mapping for it, and if so, how is it done?

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Short answer: Yes, but it won't be pretty.

Explaination: While there are algorithms to automagically UV map polygon soup, the mapping probably wont be ideal. UV mapping is an art really. Choosing where to hide the seams and where to put them to limit stretching and optimize texture space.

That said, there are lots of tools that can make manual UV mapping a heck of a lot easier. For example: http://knowledge.autodesk.com/support/maya-lt/learn-explore/caas/CloudHelp/cloudhelp/2015/ENU/MayaLT/files/Create-UVs--Automatic-Mapping-htm.html

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    $\begingroup$ Well, there are quite a bit more and quite a bit more advanced algorithms for mesh parameterization than what you linked to, so I wouldn't really subscribe to the idea that it "won't be pretty", let alone such a short answer that barely touches the tip of the iceberg. $\endgroup$ – Christian Rau Aug 14 '15 at 11:02
  • $\begingroup$ Could you elaborate/link any of those algorithms? I will be happy to add more to the answer. $\endgroup$ – RichieSams Aug 14 '15 at 11:40
  • $\begingroup$ Not at the moment really. $\endgroup$ – Christian Rau Aug 14 '15 at 13:37
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    $\begingroup$ @RichieSams: A very popular algorithm that quickly gained popularity is the least sqares conformal maps algorithm: cs.jhu.edu/~misha/Fall09/Levy02.pdf $\endgroup$ – datenwolf Aug 15 '15 at 10:53

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