We derive general results relating revivals in the dynamics of quantum many-body systems to the entanglement properties of energy eigenstates. For a lattice system of N sites initialized in a low-entangled and short-range correlated state, our results show that a perfect revival of the state after a time at most poly(N) implies the existence of "quantum many-body scars", whose number grows at least as the square root of N up to poly-logarithmic factors. These are energy eigenstates with energies placed in an equally-spaced ladder and with Rényi entanglement entropy scaling as log(N), plus an area law term for any region of the lattice. This shows that quantum many-body scars are a necessary condition for revivals, independent of particularities of the Hamiltonian leading to them, and helps explain their appearance in a number of recently found ergodicity-breaking models. We also present results for approximate revivals and for revivals of expectation values of observables.  Joint work with Henrik Wilming and Anurag Anshu