In this paper we present some new data from extension-inflation tests on a human iliac artery and then, on the basis of the nonlinear theory of elasticity, we examine a possible model to represent this data. The model considers the artery initially as a thick-walled circular cylindrical tube which may consist of two or more concentric layers. In order to take some account of the architecture (morphological structure), each layer of the material is regarded as consisting of two families of mechanically equivalent helical fibers symmetrically disposed with respect to the cylinder axis. The resulting material properties are then orthotropic in each layer. General formulas for the pressure and the axial load in the symmetric inflation of an extended tube are obtained. The starting point is the unloaded circular cylindrical configuration, but (in general unknown) residual stresses are included in the formulation. The model is illustrated by specializing firstly to the case of a single layer so that the consequences of the hypothesis of uniform circumferential stress in the physiological state can be examined theoretically. This enables the required residual stresses to be calculated explicitly. Secondly, the equations are specialized for the membrane approximation in order to show how certain important characteristics of the experimental data can be replicated using a relatively simple anisotropic membrane model.