This thesis focuses on two recent examples of non-equilibrium quantum phase transitions. In the first part we discuss discrete time crystals (DTCs), which are defined by the fact that they spontaneously break discrete time-translation symmetry. In early realizations of DTCs, many-body localization (MBL) played a crucial role in preventing the periodic drive from heating the system to infinite temperature, which would preclude any possibility of symmetry-breaking. This thesis explores the possibility that dissipation may play an equivalent role, allowing for the possibility of time-translation symmetry-breaking without MBL. We describe the results of an experiment exploring DTC order in a doped semiconductor system with significant dissipation, and a potential description of the interplay of driving, dissipation and interactions using a central spin model. In the second part we discuss measurement-induced phase transitions, where the steady-state entanglement can undergo a phase transition as a function of the measurement rate. First we explore the role of the underlying unitary dynamics in the nature of the phase transition. Previous work has revealed an apparent dichotomy between interacting and non-interacting systems, where interacting systems have a phase transition from volume-law to area-law entanglement at a finite critical measurement rate p, whereas the volume-law for non-interacting systems is destroyed at any p > 0. We study this transition for MBL systems, and find an interpolation between these extremes depending on the measurement basis. We discuss the relevance of the emergent integrability characteristic of MBL and how this intersects with the measurements. Next we study the critical properties of this transition in random 1+1D and 2+1D Clifford circuits, aiming to explore connections with percolation. We utilize a graph-state based simulation algorithm, which provides access to geometric properties of entanglement. We find bulk exponents close to percolation, but possible differences in surface behaviour
