While the transport of concentration fields arising in nature and technology is often predominantly advective, it invariably has at least a small diffusive component as well. The inclusion of diffusivity in transport studies increases their complexity significantly, as we will see.At the same time, introducing the diffusivity creates an opportunity to settle on a broadly agreeable definition for a transport barrier. Indeed, diffusive transport through a material surface is a uniquely defined, fundamental physical quantity, whose extremizing surfaces can be defined without reliance on any special notion of coherence. In the limit of zero diffusivity, the results we describe in this chapter also give a unique, physical definition of purely advective LCSs as material surfaces that will block transport most efficiently under the addition of the slightest diffusion or uncertainty to the velocity field.