We explain how we can extend Chows theorem and its generalizations 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 we apply 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.