Hodge theory provides a way to assign linear-algebraic invariants to algebraic varieties, built out of integrating holomorphic differentially forms. While studying individual Hodge structures can be difficult, understanding them in families is much more approachable. This gives natural "period maps" from algebraic varieties to moduli of Hodge structures, known as "period domains". These period domains are almost never algebraic, but they always admit a definable structure over . This makes them ideal objects for study from the perspective of complex analytic o-minimal geometry. We explain the basic theory, and give applications to some old and more recent results in Hodge theory such as the Griffiths conjecture.