(joint with B.Bakker and Y.Brunebarbe) One very fruitful way of studying complex algebraic varieties is by forgetting the underlying algebraic structure, and just thinking of them as complex analytic spaces. To this end, it is a natural and fruitful question to ask how much the complex analytic structure remembers. One very prominent result is Chows theorem, stating that any closed analytic subspace of projective space is in fact algebraic. A notable consequence of this result is that a compact complex analytic space admits at most one algebraic structure - a result which is false in the non-compact case. This was generalized and extended by Serre in his famous GAGA paper using the language of cohomology. We explain how we can extend Chows theorem and in fact all of 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. As prerequisities for this talk, it would be helpful to have gone through a first year course in Algebraic Geometry, covering in particular the theory of sheaves.