A quantum many-body system, prepared initially in a state with low entanglement, will entangle distant regions dynamically. How does this happen? I will discuss entanglement entropy growth for quantum systems subject to random unitary dynamics — i.e. Hamiltonian evolution with time-dependent noise, or a random quantum circuit. I will show how entanglement growth in this ‘noisy’ situation exhibits remarkable universal structure, which in 1D is related to the Kardar—Parisi—Zhang equation. I will argue that understanding this structure leads us to heuristic pictures for entanglement growth which may be useful for more general (non-noisy) dynamics.