Symmetry and entanglement stand as two foundational concepts in modern quantum physics that have revolutionised our understanding of quantum many-body systems. Their interplay is becoming a central theme in contemporary quantum research, influencing a broad spectrum of fields including quantum information, condensed matter theory, and high-energy physics. The MOSE project sets out to develop new theoretical frameworks to explore how entanglement and symmetries interact with each other. We will investigate various scenarios, such as the restoration of symmetries in monitored quantum circuits, the entanglement properties of supersymmetric spin chains, and the dynamics of gauge symmetries in lattice gauge theories. Our approach incorporates advanced theoretical frameworks like generalised hydrodynamics and space-time duality. We also delve into entanglement asymmetry in black hole evaporation, providing fresh perspectives on the black hole information paradox. Additionally, the project will explore the quantum and classical Mpemba effect through symmetry restoration, aiming to uncover new connections between quantum and classical dynamics. The overarching theme throughout this proposal is the monitoring of symmetries through the lens of entanglement. Finally, we will leverage the randomised measurement toolbox to examine the experimental implications of all our findings. This process will involve designing new experiments and collaborating with leading experimental groups to validate some of our theoretical predictions. This interdisciplinary effort promises to deepen our understanding of fundamental quantum phenomena and their implications for future quantum devices.