\begin{thebibliography}{ACTP16}

\bibitem[ACTP16]{ACTP}
A.~Auel, J.-L. Colliot-Th\'el\`ene, and R.~Parimala.
\newblock Universal unramified cohomology of cubic fourfolds containing a
  plane.
\newblock In {\em Brauer Groups and Obstruction Problems:\ Moduli Spaces and
  Arithmetic}, Progr. Math. Birkh\"auser, 2016.
\newblock Proceedings of a workshop at the American Institute of Mathematics.

\bibitem[Ale87]{alexeev}
V.~A. Alekseev.
\newblock On conditions for the rationality of three-folds with a pencil of del
  {P}ezzo surfaces of degree {$4$}.
\newblock {\em Mat. Zametki}, 41(5):724--730, 766, 1987.

\bibitem[AM72]{ArtMum}
M.~Artin and D.~Mumford.
\newblock Some elementary examples of unirational varieties which are not
  rational.
\newblock {\em Proc. London Math. Soc. (3)}, 25:75--95, 1972.

\bibitem[Bea77]{Beau77}
A.~Beauville.
\newblock Vari\'et\'es de {P}rym et jacobiennes interm\'ediaires.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup. (4)}, 10(3):309--391, 1977.

\bibitem[Bea86]{Beau86}
A.~Beauville.
\newblock Le groupe de monodromie des familles universelles d'hypersurfaces et
  d'intersections compl\`etes.
\newblock In {\em Complex analysis and algebraic geometry ({G}\"ottingen,
  1985)}, volume 1194 of {\em Lecture Notes in Math.}, pages 8--18. Springer,
  Berlin, 1986.

\bibitem[Bea16a]{beau-survey}
A.~Beauville.
\newblock The {L}\"uroth problem.
\newblock In {\em Rationality Problems in Algebraic Geometry}, volume 2172 of
  {\em Lecture Notes in Math.} Springer, Cham; Fondazione C.I.M.E., Florence,
  2016.
\newblock {\tt arXiv:1507.02476}.

\bibitem[Bea16b]{beau-6}
A.~Beauville.
\newblock A very general sextic double solid is not stably rational.
\newblock {\em Bull. Lond. Math. Soc.}, 48(2):321--324, 2016.

\bibitem[Che06]{Cheltsov-NNQT}
I.~Cheltsov.
\newblock Nonrational nodal quartic threefolds.
\newblock {\em Pacific J. Math.}, 226(1):65--81, 2006.

\bibitem[CTP16a]{ct-pir-cyclic}
J.-L. Colliot-Th{\'e}l{\`e}ne and A.~Pirutka.
\newblock Cyclic covers that are not stably rational.
\newblock {\em Izv. Ross. Akad. Nauk Ser. Mat.}, 80(4):35--48, 2016.

\bibitem[CTP16b]{ct-pirutka}
J.-L. Colliot-Th{\'e}l{\`e}ne and A.~Pirutka.
\newblock Hypersurfaces quartiques de dimension 3: non-rationalit\'e stable.
\newblock {\em Ann. Sci. \'Ec. Norm. Sup\'er. (4)}, 49(2):371--397, 2016.

\bibitem[Gro71]{SGAI}
{\em Rev\^etements \'etales et groupe fondamental}.
\newblock Lecture Notes in Mathematics, Vol. 224. Springer-Verlag, Berlin-New
  York, 1971.
\newblock S{\'e}minaire de G{\'e}om{\'e}trie Alg{\'e}brique du Bois Marie
  1960--1961 (SGA 1), Dirig{\'e} par Alexandre Grothendieck. Augment{\'e} de
  deux expos{\'e}s de M. Raynaud,.

\bibitem[HKT16a]{HKT}
B.~Hassett, A.~Kresch, and Y.~Tschinkel.
\newblock On the moduli of degree $4$ {D}el {P}ezzo surfaces.
\newblock In {\em {D}evelopment of {M}oduli {T}heory}, volume~69 of {\em
  Advanced Studies in Pure Mathematics}, pages 349--386. Mathematical Society
  of Japan, Tokyo, 2016.

\bibitem[HKT16b]{HKT-conic}
B.~Hassett, A.~Kresch, and Y.~Tschinkel.
\newblock Stable rationality and conic bundles.
\newblock {\em Math. Ann.}, 365(3-4):1201--1217, 2016.

\bibitem[HT14]{HTCEJM}
B.~Hassett and Y.~Tschinkel.
\newblock Quartic del {P}ezzo surfaces over function fields of curves.
\newblock {\em Cent. Eur. J. Math.}, 12(3):395--420, 2014.

\bibitem[HVA13]{HaVA}
B.~Hassett and A.~V{\'a}rilly-Alvarado.
\newblock Failure of the {H}asse principle on general {$K3$} surfaces.
\newblock {\em J. Inst. Math. Jussieu}, 12(4):853--877, 2013.

\bibitem[IM00]{IlMark}
A.~Iliev and D.~Markushevich.
\newblock The {A}bel-{J}acobi map for a cubic threefold and periods of {F}ano
  threefolds of degree 14.
\newblock {\em Doc. Math.}, 5:23--47 (electronic), 2000.

\bibitem[IP99]{isk-prokhorov}
V.~A. Iskovskikh and Y.~G. Prokhorov.
\newblock Fano varieties.
\newblock In {\em Algebraic geometry, {V}}, volume~47 of {\em Encyclopaedia
  Math. Sci.}, pages 1--247. Springer, Berlin, 1999.

\bibitem[MM82]{MoriMukai-manuscripta}
S.~Mori and S.~Mukai.
\newblock Classification of {F}ano {$3$}-folds with {$B_{2}\geq 2$}.
\newblock {\em Manuscripta Math.}, 36(2):147--162, 1981/82.

\bibitem[Sar82]{Sar}
V.~G. Sarkisov.
\newblock On conic bundle structures.
\newblock {\em Izv. Akad. Nauk SSSR Ser. Mat.}, 46(2):371--408, 432, 1982.

\bibitem[Tot16]{totaro-JAMS}
B.~Totaro.
\newblock Hypersurfaces that are not stably rational.
\newblock {\em J. Amer. Math. Soc.}, 29(3):883--891, 2016.

\bibitem[vG05]{vG}
B.~van Geemen.
\newblock Some remarks on {B}rauer groups of {$K3$} surfaces.
\newblock {\em Adv. Math.}, 197(1):222--247, 2005.

\bibitem[Voi14]{VoisinJEMS}
C.~Voisin.
\newblock On the universal {$\CH_0$} group of cubic hypersurfaces.
\newblock {\em J. Eur. Math. Soc. (JEMS)}, to appear, 2014.
\newblock {\tt arXiv:1407.7261v2}.

\bibitem[Voi15]{Voisin}
C.~Voisin.
\newblock Unirational threefolds with no universal codimension {$2$} cycle.
\newblock {\em Invent. Math.}, 201(1):207--237, 2015.

\end{thebibliography}