The coniveau of a variety over the complex numbers  is the maximal codimension of an algebraic subset supporting its transcendental cohomology. On one hand, the coniveau is conjecturally predicted by the  vanishing of Hodge numbers. On the other hand, it conjecturally predicts the vanishing of certain Chow groups of cycles cohomologous to 0. I will describe the  various notions, how they relate to refined decomposition of the diagonal, and what is known in the case of hypersurfaces or complete intersections.