A sheet of paper is modelled as a network of cellulose fibres, either cylindrical or band-shaped. The equations for creeping flow through such structures are solved, and the calculated permeabilities are compared with measured values. Flow through some paper structures such as pulp sheets and handsheets of unbeaten sulphate pulp is adequately described by the structural model involving band-shaped fibres when a fibre aspect ratio of 3.5 is used. For newsprint sheets the measured permeability is lower than that predicted by the models when physically realistic values of the aspect ratio are taken. A total of 19 different paper grades have been characterised by measurement of the total specific surface area and the fibre orientation ratio in addition to the measurements of effective diffusivity, permeability and porosity. Permeability and effective diffusivity correlate with each other and permeability correlates with fibre orientation, so that at constant porosity, permeability decreases with increasing fibre orientation.