A Hodge structure is a certain linear algebraic datum.  Importantly, the cohomology groups of any smooth projective algebraic variety come equipped with Hodge structures which encode the integrals of algebraic differential forms over topological cycles.  In these lectures, we will slowly introduce the definition of a Hodge structure, and proceed to describe their parameter spaces and how they vary in algebraic families.  Throughout we will emphasize how the analytic structures that show up are all definable in an o-minimal structure, and how this bridges the gap between transcendental operations (like integration) and the algebraic structures on the underlying varieties.