Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. For integrable models, novel hydrodynamic approaches based on a quasiparticle picture emerged as a new platform allowing for a quantitative understanding of quantum information dynamics in quantum many-body systems. Remarkably, this gives fresh insights on how thermodynamics emerges in isolated out-of-equilibrium quantum systems.

I will start by reviewing this new unifying framework. I will then discuss several applications to entanglement-related quantities, such as entanglement entropies, mutual information, logarithmic negativity. I will also show how the framework allows to study the interplay between quantum information dynamics and transport of local conserved quantities. Finally, I will derive some simple bounds on the quantum information scrambling in out-of-equilibrium systems.