(joint with B.Bakker and Y.Brunebarbe)
We explain how we can extend Chows theorem and GAGA to the non-compact case by working with complex analytic structures that are "tame" in the precise sense defined by o-minimality. This leads to some very general "algebraization" theorems, which can be used to obtain new results in Hodge Theory. In particular, we use this technology to prove a conjecture of Griffiths on the algebraicity and quasi-projectivity of images of period maps.