The period matrix of a curve effectively describes how the complex structure varies; this is Torelli's theorem dating from the beginning of the nineteenth century. In the 1950s during the first revolution of algebraic geometry, attention shifted to higher dimensions and one of the guiding conjectures, the Hodge conjecture got formulated. In the late 1960s and 1970s Griffiths, in an attempt to solve this conjecture, generalized the classical period matrices introducing period domains and period maps for higher dimensional manifolds. He then found some unexpected new phenomena for cycles on higher dimensional algebraic varieties, which were later made much more precise by Clemens, Voisin, Green and others. This book presents this development starting at the beginning: the elliptic curve. This and subsequent examples (curves of higher genus, double planes) are used to motivate the concepts that play a role in the rest of the book. The basic theory of the period map as developed by Griffiths is developed in the first part of the book. Then, in the second part spectral sequences and Koszul complexes are introduced and used to derive the recent results related to cycles just alluded to.
Finally, in the third part differential geometric methods are explained leading up to proofs of Arakelov-type theorems, the theorem of the fixed part, the rigidity theorem, etc. In this part Higgs bundles are discussed and their relation to harmonic maps; this leads to striking results such as the fact that compact quotients of certain period domains can never admit a Kahler metric or that certain lattices in classical Lie groups can't occur as the fundamental group of a Kahler manifold.