  We carry out a complete birational classification of the degree g universal
Jacobian P_g over the moduli space of curves, highlighting the transition cases
g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira
dimension zero for g=10 and Kodaira dimension 19 for g=11. For g>11, the
variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by
Iitaka's easy addition formula for fibre spaces. In particular, we disprove the
expectation that P_g and M_g have the same Kodaira dimension for all genera.
