We explain the mechanism leading to directed chaotic transport in Hamiltoni
an systems with spatial and temporal periodicity. We show that a mixed phas
e space comprising both regular and chaotic motion is required and we deriv
e a classical sum rule which allows one to predict the chaotic transport ve
locity from properties of regular phase-space components. Transport in quan
tum Hamiltonian ratchets arises by the same mechanism as long as uncertaint
y allows one to resolve the classical phase-space structure. We derive a qu
antum sum rule analogous to the classical one, based on the relation betwee
n quantum transport and band structure.