We develop birational versions of Voevodsky's triangulated categories of
motives over a field, and relate them with the pure birational motives studied
in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified
cohomology in this framework, leading to "higher derived functors of unramified
cohomology".Comment: Compared to the initial version: previous Subsection 4.2 has been
upgraded to Section 5; previous Lemmas 5.2.5 and 5.2.6 have been corrected to
Proposition 6.2.5 and Lemma 6.2.6; at the referee's request, previous
Appendix B and the proof of previous Proposition C.1.1 (now A.4.1) have been
removed (please consult the initial version for them
