The eigenstate entanglement entropy is a powerful tool to distinguish integrable from generic quantum-chaotic Hamiltonians. In integrable models, the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) has a volume-law coefficient that generally depends on the subsystem fraction. In contrast, the volume-law coefficient is maximal (subsystem fraction independent) in quantum-chaotic models. In the seminar I will present an overview of our current understanding of bipartite entanglement entropies in many-body quantum states above the ground states, and contrast analytical predictions with numerical results for eigenstates of physical Hamiltonians.