The classifcation of algebraic varieties is a central part of algebraic geometry. Among the common tools that are employed are birational geometry, especially using the analysis of singularities, and Hodge theory. The latter is frequently used as a sort of "black box": the cohomology of an algebraic variety has a functorial mixed Hodge structure and the formal properties of the corresponding category have very strong consequences. On the other hand, geometric structures arise naturally in Hodge theory and one aspect of this will be the focus of these lectures. This aspect originates from the fact that mixed Hodge structures have extension data expressed by linear algebra.