The extensive field of driven quantum systems has seen numerous theoretical and experimental advances in recent years, with studies ranging from measurement techniques, over engineering of out-of-equilibrium phases of matter to considerations of active matter on the quantum level. In this thesis, we explore three timely directions of this body of research. First, we consider arrays of Josephson junctions, which have become highly tunable in experiments due to recent fabricational progress. Numerical calculations suggest that transverse magnetic fields induce a cascade of different vortex lattice states on the arrays. We find in this thesis that the measurement of the full resistivity tensor gives direct access to information about the vortex lattices. Motivated by a recent experiment, we further study an array of $ϕ_0$-junctions. We find that going beyond nearest neighbor coupling, the $ϕ_0$-shifts lead to an asymmetric pinning potential for the vortices which consequently facilitates a Josephson diode effect via vortex depinning. Second, we turn towards periodically driven spin chains. Motivated by a recent experiment on a quantum processor, we study the Floquet quantum Ising model in its fermionic formulation. Depending on the parameters, the model can host edge modes that are associated with a topological, equilibrium (Majorana zero modes) or out-of-equilibrium (Majorana pi modes) phase. In contrast to the equilibrium phase, we find that the edge modes in the out-of-equilibrium phase are characterized by remarkable resistance against parity-breaking perturbations, which we explain in terms of a stroboscopic perturbation theory. Following this, we study spin correlation functions of a Floquet time crystal. Here we find a connection of pairings in the spectrum of the Floquet operator to the defining period-2 spin correlations of the time-crystalline phase. Specifically, we can express the correlation functions in terms of the Fourier transform of the pairing distribution, which is approximately of log-normal form. We also discuss implications for the phase diagram of the time crystal. The third and final topic of this thesis is the emerging field of active quantum matter. We first state the general framework for a single active quantum particle, after which we consider the specific example of a spin-1/2 particle. We then show under which conditions the particle carries out active motion and demonstrate that the motion is characterized by genuine quantum features.
