We consider an axisymmetric polymeric membrane inflated by a uniform pressure difference acting across the membrane. The polymeric material is described by an arbitrary combination of a viscoelastic and a purely viscous component to the stress. Some viscoelastic materials described by a Mooney-Rivlin model show a monotone increasing pressure during inflation of a spherical membrane. These materials develop a homogeneous membrane thickness in agreement with the Considere-Pearson condition. Molecularly based models such as the neo-Hookean, Doi-Edwards or Tom-Pom model show a pressure maximum when inflated. Membranes described by these models develop a local thinning of the membrane which may lead to bursting in finite time.