jats:p For the first time, the problem of the inflation of a jats:italicnonlinear viscoelastic</jats:italic> thick-walled spherical shell is considered. Specifically, the wall has jats:italicquasilinear viscoelastic</jats:italic> constitutive behaviour, which is of fundamental importance in a wide range of applications, particularly in the context of biological systems such as hollow viscera, including the lungs and bladder. Experiments are performed to demonstrate the efficacy of the model in fitting relaxation tests associated with the volumetric inflation of jats:italicmurine bladders</jats:italic> . While the associated jats:italicnonlinear elastic</jats:italic> problem of inflation of a balloon has been studied extensively, there is a paucity of studies considering the equivalent jats:italicnonlinear viscoelastic</jats:italic> case. We show that, in contrast to the elastic scenario, the peak pressure associated with the inflation of a neo-Hookean balloon is jats:italicnot</jats:italic> independent of the shear modulus of the medium. Moreover, a novel numerical technique is described in order to solve the jats:italicnonlinear Volterra integral equation</jats:italic> in space and time originating from the fundamental problem of inflation and deflation of a thick-walled nonlinear viscoelastic shell under imposed pressure. </jats:p>