We show that in a medium consisting of asymmetric cells, neutrons can “drift,” or diffuse, in a special preferred direction. The drift is caused by selective asymmetric changes in the cross sections in each cell. We describe several physical mechanisms that produce a drift, and we briefly discuss a possible application in a reflector design. (A reflector constructed of asymmetric cells, oriented so that the drift is always directed toward the reactor core, would be more efficient than a homogeneous driftless reflector.) Our theoretical treatment consists of an asymptotic analysis of the one-dimensional neutron transport equation. We show that a simple modification of the diffusion equation describes the neutron drift, and we provide numerical results for several problems. We also numerically compare the solution of an initial value problem for the transport equation in an asymmetric cellular medium to the corresponding diffusion theory problem. The results are in reasonably good agreement for both short and long times.