It is well-known that the Hilbert space of a quantum many-body system grows exponentially with the number of particles in the system. Drive the system out of equilibrium so that the degrees of freedom are now dynamic and the result is an extremely complicated problem. With that comes a vast landscape of new physics, which we are just recently starting to explore. In this proposal, we study the dynam- ics of two paradigmatic classes of quantum many-body systems: quantum chaotic and integrable systems. We leverage certain tools commonly employed in equilibrium many-body physics, as well as others tailored to the realm of non-equilibrium scenar- ios, in order to address various problems within this evolving field. Our contributions are the following: Inspired by random matrix theory and random unitary circuits subject to projective measurements, we first uncover a novel phase transition in a model of random tensor networks separating an area-law from a logarithmic-law in the scaling of entanglement entropy of a many-body wavefunction. Next, we study transport in the Rule 54 cellular automaton, a paradigmatic integrable model displaying just two species of solitons of different chiralities. Our contribution here is a sound numerical verification of some of the formulas for transport coefficients recently derived within a generalized hydrodynamic approach valid for integrable systems. Using the equations of generalized hydrodynamics as a starting point we then propose a new phenomenological scheme based on a relaxation-time approximation widely used in kinetics, but fundamentally different, to study the experimentally relevant regime where only a few conservation laws are present. We then aim at uncovering the hydrodynamics of integrability-breaking starting from fully microscopic dynamics. To do so we study a noisy version of the Rule 54 model and of the hard-rod gas, where the source of noise in both models is backscattering of solitons. We find that these models of integrability-breaking are atypical in that in the former relaxation occurs at long time scales owing to the presence of kinetic constraints, and the latter displays singular transport signatures as a result of infinitely many conserved charges despite the model being nonintegrable. Finally, we conclude by studying operator spreading in both integrable and chaotic quantum chains. Using hydrodynamics and tensor net- work simulations we find distinctive signatures of these two classes of models when looking at their operator front.