The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.

Chapter 1 begins with an introduction to the principle of universality, seen through the lens of Landau's theory of symmetry-breaking order parameters and phase transitions. It then covers the paradigmatic understanding of equilibrium universality via the set of ideas known as the renormalization group. Moving out of equilibrium, we discuss Floquet theory and periodically driven systems, before finally mentioning other examples of non-equilibrium universality as context for our later results.

Chapter 2 investigates quantum critical systems described by a conformal field theory (CFT) driven at their boundary, both periodically in time and via noise. In the time-periodic case, we find that the system displays multiple dynamical regimes depending on the drive frequency. We compute the behavior of quantities including the entanglement entropy and Loschmidt echo, confirming analytic predictions from field theory by exact numerics on the transverse field Ising model, and demonstrate universality by adding non-integrable perturbations. The dynamics naturally separate into three regimes: a slow-driving limit, which has an interpretation as multiple quantum quenches with amplitude corrections from CFT; a fast-driving limit, in which the system behaves as though subject to a single quantum quench; and a crossover regime displaying heating. The universal Floquet dynamics in all regimes can be understood using a combination of boundary CFT and Kibble-Zurek scaling arguments. We then move on to stochastic boundary driving. We formulate a generic ansatz for the dynamical scaling form of the typical Loschmidt echo and corroborate it with exact numerical calculations in the case of a spin impurity driven by shot noise in a quantum Ising chain. We find that due to rare events the dynamics of the mean echo can follow very different dynamical scaling than the typical echo for certain classes of drives. Our results are insensitive to irrelevant perturbations of the bulk critical model and apply to all the microscopic models in the same universality class.

Chapter 3 considers disordered driven quantum systems, in particular, periodically driven (Floquet) systems that undergo many-body localization (MBL). We study transitions between distinct phases of one-dimensional Floquet systems. We argue that these are generically controlled by infinite-randomness fixed points of a strong-disorder renormalization group procedure. Working in the fermionic representation of the prototypical Floquet Ising chain, we leverage infinite randomness physics to provide a simple description of Floquet (multi)criticality in terms of a new type of domain wall associated with time-translational symmetry-breaking and the formation of `Floquet time crystals'. We validate our analysis via numerical simulations of free-fermion models sufficient to capture the critical physics. We then introduce a real-space renormalization group approach for Floquet MBL systems, asymptotically exact in the strong-disorder limit, and exemplify its use on the periodically driven interacting quantum Ising model. We analyze the universal physics near the critical lines and multicritical point of this model, and demonstrate the robustness of our results to the inclusion of weak interactions.

Chapter 4 pivots to consider quantum effects in the context of hydrodynamics, in particular examining the phenomenon of Coulomb drag as a quantum analogue of the shear viscosity. Two conducting quantum systems coupled only via interactions can exhibit the phenomenon of Coulomb drag, in which a current passed through one layer can pull a current along in the other. As a transport signature, Coulomb drag has been found to be a sensitive probe of the systems' microscopic structure and displays a rich dependence with temperature. However, in quantum systems with particle-hole symmetry -- for instance, the half-filled Hubbard model or graphene near the Dirac point -- the Coulomb drag effect is vanishingly small. We point out that its thermal analogue, whereby a thermal current in one layer pulls a thermal current in the other, is nonzero at these particle-hole symmetric points and is indeed the dominant form of drag in particle-hole symmetric systems. By studying a quantum quench in the paradigmatic one-dimensional Hubbard model, we show that thermal drag, in marked contrast to charge drag, displays a non-Fermi's Golden Rule growth at short times due to a logarithmic scattering singularity generic to one dimension. Exploiting the integrability of the Hubbard model, we obtain the long-time limit of the quench for weak interactions. Finally, we connect these results to a thermal drag conductivity via an appropriate Kubo formula, and comment on thermal drag effects in two-dimensional systems.

In Chapter 5, we close with comments on future lines of investigation, and on likely routes to a full understanding of non-equilibrium universality.