While dissipation is in general perceived as a destructive feature of a quantum system, it can also be utilized to engineer nontrivial states, often in conjunction with pushing a system out of equilibrium. We study various non-equilibrium phases in an ultracold quantum gas coupled to a high-finesse optical cavity. We utilize both effective models, such as the three-level Dicke model, and numerical methods to characterize the emerging phases ranging from time crystals to dark states. We put forth the first experimental realization of a limit cycle in an atom-cavity system and explain its origin using bifurcation theory, which also enables us to understand the system's route to aperiodic dynamics. Finally, we propose to use the atom-cavity system to create a quantum rotational sensor not only for static high precision frequency measurements but also for inertial measurements and navigation.