In the first part of the talk, I consider the non-equilibrium time evolution of piecewise homogeneous states in the XXZ spin-1/2 chain.
I overview the solution to the dynamics in the limit of large time and discuss some remarkable properties of the profiles of charges and currents.
In the second part of the talk, I describe an unusual mechanism of prethermalization based on the presence of a symmetry of the pre-quench Hamiltonian which is spontaneously broken at zero temperature and is explicitly broken by the post-quench Hamiltonian.
 B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016).
 V. Alba and M. Fagotti, Prethermalization at low temperature: the scent of a spontaneously broken symmetry, arXiv:1701.05552 (2017)