Intermediate Jacobians and Abel-Jacobi maps play a central role in Hodge theory, with many applications in complex algebraic geometry to moduli theory and rationality questions. In this talk I will discuss joint work with Jeff Achter and Charles Vial showing how to use these techniques over any perfect field, answering some natural questions, such as whether the intermediate Jacobian and Abel-Jacobi map of a given smooth projective variety descend to the field of definition of the variety. Other applications include a special case of a conjecture of Orlov regarding derived categories of threefolds.