A fundamental tool to study algebraic varieties is given by morphisms to projective space. A line bundle is called semiample if some positive tensor multiple provides a morphism to projective space. Unlike the case of ample line bundles, there are no general numerical criteria to determine whether a line bundle is semiample. In particular, proving the semiampleness of line bundles can be a challenging task. In this talk, we will report on recent developments regarding the semiampleness of line bundles coming from Hodge theory. As a consequence, we will obtain functorial compactifications of general period images, generalizing classic work of Baily--Borel and thus confirming a conjecture of Griffiths. This talk is based on joint work with Bakker, Mauri, and Tsimerman.