A quantum many-body system, prepared initially in a state with low entanglement, will entangle distant regions dynamically. While entanglement has served a crucial role in characterizing quantum many-body systems, very few general heuristics exist for understanding its dynamics. We discuss the growth of the entanglement entropy for quantum systems subject to random, local unitary evolution -- i.e. Hamiltonian evolution with time-dependent noise, or a random quantum circuit. Entanglement growth in this "noisy" situation exhibits a remarkable universal structure, which in 1D is related to the Kardar-Parisi-Zhang equation. Understanding entanglement growth for random dynamics leads to heuristic pictures that apply to more general (i.e. non-noisy) dynamics, both in 1D and in higher dimensions. We also discuss exact results for the growth of local operators in this "noisy" environment, and the implications for scrambling in random quantum circuits.