The long-time asymptotics of thermodynamically large systems is an important problem in non-equilibrium statistical mechanics. The thermal equilibrium states and their behaviour under local perturbations are well understood for both classical and quantum systems. Non-equilibrium steady states play a central role in the description of physical processes far from equilibrium. On the other hand, the mechanism of relaxation to these states and their local properties and stability are far less studied. The main goal of this project is to investigate these problems from the mathematical point of view in the framework of various physically relevant models. In the context of classical systems, our main focus will be on the motion of particles in the flow of 2D incompressible viscous fluid and coupled Hamiltonian systems. While these two problems are rather different from the mathematical point of view, they are very close conceptually: large systems, staying in a non-equilibrium steady state in the course of the time, interact with a small one and cause it to stabilise as time goes to infinity. The quantum side of the project splits into three different, but intimately related directions. Quantum walkers, arising as an approximation for more realistic systems, the analysis of the full statistics of the particle currents and investigation of mean field type regimes form our first group of problems. The understanding of local properties and stability of non-equilibrium steady states are fundamental questions in statistical mechanics. We shall investigate the concept of local temperature in various physical models and contexts. Finally, we shall study the concept of entanglement entropy used in quantum physics and in quantum information theory, with the aim to provide a rigorous interpretation in the context of models of quantum field theory.