Transport in strongly quantum systems is challenging to understand. I will describe a recently obtained bound on transport in terms of a characteristic quantum velocity (the Lieb-Robinson velocity) and the local thermalization time. This bound sheds some light on experiments in both condensed matter systems and ultracold atomic gases. At finite temperatures, a more powerful velocity is the so-called butterfly velocity, that is intimately related to quantum chaos. This velocity is still poorly understood; I will present some forthcoming results that constrain the temperature dependence of the butterfly velocity in terms of the underlying quantum scrambling of the system.