A big classification result that I'm aware of is due to Gromov and Lawson.

>**Theorem.** Let $M$ be a compact simply connected manifold of dimension $\geq 5$, which is not a spin. Then $M$ admits a metric of positive scalar curvature. A spin manifold of dimension $\geq 5$ carries a metric of positive scalar curvature iff its $\alpha$-genus is zero.

The Seiberg-Witten invariants provide special obstructions to existence of a metric of positive scalar curvature  in dimension 4.


There are two good survey articles on the subject by J. Rosenberg ([link 1][1]) and S. Stolz ([link 2][2]). 

 


  [1]: http://www.ams.org/mathscinet/pdf/2408269.pdf?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=&s5=Manifolds%20of%20positive%20scalar%20curvature&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1
  [2]: http://www.ams.org/mathscinet/pdf/1937026.pdf?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=&s5=Manifolds%20of%20positive%20scalar%20curvature&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2