The paper proposes a covariant generalization of the holographic entanglement entropy proposal to understand the time-dependent behavior of entanglement entropy in quantum field theories (QFTs). The authors motivate their construction by considering the analogy between spacelike geodesics in Euclidean spacetimes and extremal surfaces in Lorentzian spacetimes. They introduce the concept of light-sheets, which are spacelike surfaces in Lorentzian spacetimes that bound the entropy passing through their light cones. The area of an extremal surface anchored at a specified region on the boundary in the AdS/CFT correspondence is proposed as the entanglement entropy of that region. The authors demonstrate that this construction reduces to the minimal surface prescription for static spacetimes and show its consistency through various examples, including rotating black holes and gravitational collapse. They also discuss the interpretation of Euclidean wormhole geometries with multiple boundaries as states in a non-interacting but entangled set of QFTs. The paper provides a detailed mathematical framework and physical insights into the covariant holographic entanglement entropy, offering a new tool for understanding the dynamics of entanglement entropy in time-dependent QFTs.The paper proposes a covariant generalization of the holographic entanglement entropy proposal to understand the time-dependent behavior of entanglement entropy in quantum field theories (QFTs). The authors motivate their construction by considering the analogy between spacelike geodesics in Euclidean spacetimes and extremal surfaces in Lorentzian spacetimes. They introduce the concept of light-sheets, which are spacelike surfaces in Lorentzian spacetimes that bound the entropy passing through their light cones. The area of an extremal surface anchored at a specified region on the boundary in the AdS/CFT correspondence is proposed as the entanglement entropy of that region. The authors demonstrate that this construction reduces to the minimal surface prescription for static spacetimes and show its consistency through various examples, including rotating black holes and gravitational collapse. They also discuss the interpretation of Euclidean wormhole geometries with multiple boundaries as states in a non-interacting but entangled set of QFTs. The paper provides a detailed mathematical framework and physical insights into the covariant holographic entanglement entropy, offering a new tool for understanding the dynamics of entanglement entropy in time-dependent QFTs.