  We begin by introducing the concept of a Hodge structure and give some of its
basic properties, including the Hodge and Lefschetz decompositions. We then
define the period map, which relates families of Kahler manifolds to the
families of Hodge structures defined on their cohomology, and discuss its
properties. This will lead us to the more general definition of a variation of
Hodge structure and the Gauss-Manin connection. We then review the basics about
mixed Hodge structures with a view towards degenerations of Hodge structures;
including the canonical extension of a vector bundle with connection, Schmid's
limiting mixed Hodge structure and Steenbrink's work in the geometric setting.
Finally, we give an outlook about Hodge theory in the Gross-Siebert program.
