This book is about Prof. Claire Voisin's lectures delivered for the "Lagrange Chair" at the University 'La Sapienza' in Rome during the fall of 1996. Their subject is dominated by mirror symmetry, which gives conjecturally a new way of describing the periods of a Calabi-Yau threefold (and eventually n-fold). We first present the variations of Hodge structure of Calabi-Yau threefolds in a "mirror symmetry independent" point of view, and then mirror symmetry. The reason for doing this is that these variations of Hodge structure proved to be quite special from the point of view of differential geometry and of algebraic geometry and this is not apparent from the mirror symmetry approach. Also, it may be that some algebro-geometric aspects could play a role in a better understanding of mirror symmetry.