P=NP: What are the most promising methods?

Question

I know that P=NP has not been solved up to now, but can anybody tell me something about the following: What are currently the most promising mathematical / computer scientific methods that could be helpful to tackle this problem? Or are there even none such methods known to be potentially helpful up to now? Is there any (free) compendium on this topic where I can find all / most of the research done in this area?

Answer 1

An excellent overview appeared last year in Communications of the ACM. I think it became the most downloaded article of CACM ever, so your question may be relevant after all :-)

The Status of the P=NP Problem, Lance Fortnow, Communications of the ACM, Vol. 52 No. 9, 2009