This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, which is observed once the bubble exceeds a certain critical volume. In this paper, the employment of some choice modelling assumptions allows insights into the mechanisms of the jump discontinuity which cannot be ascertained experimentally. The ambient fluid is assumed incompressible and the flow irrotational, with viscoelastic effects included through the stress balance on the bubble surface. The governing equations are solved using the boundary element method. Some Newtonian predictions are discussed before investigating the role of viscoelasticity. The model predicts the trademark cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies. (C) 2010 Elsevier B.V. All rights reserved.