Abstract: A global quench is an interesting setting where we can study thermalization of subsystems in a pure state. We investigate the spread of entanglement following a global quench in free field theory, holography, spin chains, and toy models. We find that the results in free scalar field theory are exactly reproduced by the model of free streaming quasiparticles, but strongly interacting chaotic systems spread entanglement more effectively. Based on recent insight into many-body quantum chaos, we propose two inequalities on the entropy, and suggest that the true evolution of the entropy is well approximated by saturating the combined bounds. A model based on the chaotic growth of operators saturates the combined bounds, and is a promising candidate to replace the quasiparticle model for chaotic systems.