  We discuss various results and questions around the Grothendieck period
conjecture, which is a counterpart, concerning the de Rham-Betti realization of
algebraic varieties over number fields, of the classical conjectures of Hodge
and Tate. These results give new evidence towards the conjectures of
Grothendieck and Kontsevich-Zagier concerning transcendence properties of the
torsors of periods of varieties over number fields.
  We notably establish that the Grothendieck period conjecture holds in degree
1 for products of curves, of abelian varieties, and of K3 surfaces, and that it
holds in degree 2 for smooth cubic fourfolds.
