Hodge theory for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the Jacobian variety. Hodge extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties. In this expository paper, some of these developments are reviewed, with special emphasis on instances where these extensions can be used to study the geometry – especially the algebraic cycles – on the variety.
