  A classification of partially hyperbolic diffeomorphisms on 3-dimensional
manifolds with (virtually) solvable fundamental group is obtained. If such a
diffeomorphism does not admit a periodic attracting or repelling
two-dimensional torus, it is dynamically coherent and leaf conjugate to a known
algebraic example. This classification includes manifolds which support Anosov
flows, and it confirms conjectures by Rodriguez Hertz--Rodriguez Hertz--Ures
and Pujals in the specific case of solvable fundamental group.
