The universal Jacobian J_{g} is the fibration over the moduli space of curves M_{g} whose fibers are the Jacobian varieties of curves of genus g. I will report on joint work with A. Verra in which we carry out a complete birational classification of J_{g} by Kodaira dimension. For g < 10 the universal Jacobian is a unirational variety, for g > 11 it has Kodaira dimension 3g − 3, whereas in the transition cases g = 10, 11 it behaves ”exotically” and has Kodaira dimension 0 and 19 respectively.