A sheet of paper is modelled as a two-dimensional network of cellulose fibres. The fibres are assumed to be either cylindrical or band-shaped. Both well-ordered fibrous structures and fibrous structures in which a random arrangement of fibres is assumed are studied. The equations for creeping flow through such structures are solved, and the calculated permeabilities are compared with measured values. Flow through such paper structures as pulp sheets and handsheets of unbeaten sulphate pulp is found to be adequately described by a structural model that assumes cellulose fibres to be band-shaped when a fibre aspect ratio of 3.5 (well-ordered structure) or 5 (random fibre distribution) is used. These values of the fibre aspect ratio compare favourably with the values used when gas diffusion through the same sheets is modelled. For newsprint sheets the measured permeability is found to be lower than that predicted by the models when physically realistic values for the aspect ratio are taken. It is also found that for all the pulp and paper grades investigated (a total of 19) the measured permeabilities and effective diffusivities correlate with each other.