Hodge theory provides the notions of "Hodge structure", developed by Griffiths, and "mixed Hodge structure" introduced by Deligne. It is a powerful tool for studying the topology of (families of) complex algebraic varieties, and this lecture presents the main results, as well as some recent applications. There are three different aspects to Hodge theory. The first aspect, briefly mentioned in this lecture, is the use of analytic methods (harmonic forms) to show the Hodge decomposition theorem, the ddbar lemma and cancellation theorems, as well as the difficult Lefschetz theorem. The results are accepted as the starting point of the theory. The second aspect, developed by Deligne, is then purely formal: it consists in identifying and exploiting the properties of the category of polarized Hodge structures and that of the category of mixed Hodge structures in order to show profound results on the topology of algebraic varieties and their families.
The third fundamental aspect of the field is the interaction between transcendental data (topology) and algebraic data (algebraic cycles). Hodge theory, i.e. knowledge of Hodge structures, or even just Hodge numbers, conjecturally gives us a way of measuring the coniveau, i.e. the codimension of the support of the transcendental part of the cohomology. According to Bloch and Beilinson, this could also be calculated via the study of Chow groups. The third part of this lecture explains how these conjectures are formulated via the notion of "diagonal decomposition" and describes some recent progress.