This doctoral thesis analytically and numerically examines some of the most important concepts in quantum correlations in low-dimensional physics: entanglement and out-of-equilibrium dynamics.As John Bell once said: "Entanglement expresses the spooky nonlocality inherent to quantum mechanics", and its study not only concerns the foundations of any quantum theory, but also has important applications in quantum information and condensed matter theory, amongst others.The first chapters are devoted to the study of "entanglement entropies", a popular measure of the "quantumness" of a physical system. The main focus of the analysis is the one-dimensional XYZ spin-1/2 chain in equilibrium, an interacting theory which in addition to being integrable also has interesting scaling limits, such as the sine-Gordon field theory.Moving away from equilibrium the subsequent chapters deal with the dynamics of quantum correlators after an instantaneous quantum quench. The emphasis is on two different models and techniques; the transverse field Ising chain is studied using the form-factor approach and the O(3) non-linear sigma model is studied by means of the semi-classical theory.In the final chapter the author highlights an important general result: in the absence of long-range interactions in the final Hamiltonian the dynamics of a quantum system are determined by the same statistical ensemble that describes static correlations.
