This paper addresses single- and two-phase flows in channels having widths in the range 0.1-10 microns. The focus is on differences in phenomena compared to mixing and flow in macro-channels. Major differences include the speed of cross-channel diffusion, the importance of axial diffusion, and effects due to the finite thickness of the interface between partially miscible phases and the chemical nature of the channel walls. Results and computational techniques are provided for estimating mixing times in initially striated flows, for the effects of combined cross-channel and axial diffusion on chemical reactions, and on the spatial-mean concentrations, convected-mean concentrations and relative flow rates of partially miscible components and the possibility of flow enhancement through the choice of wall chemistry.