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Bubbles rising in viscoelastic liquids may exhibit a jump discontinuity of the rise velocity as a critical bubble volume is exceeded. This phenomenon has been extensively investigated in the literature, both by means of experiments and via numerical simulations. The occurrence of the velocity jump has been associated with a change of the bubble shape, accompanied by the formation of a pointed tip at the rear end and to the appearance of a so-called negative wake, with the liquid velocity behind the bubble pointing in a direction opposite to that in Newtonian fluids. We revisit this topic, starting with a review of the state of knowledge on the interrelations between the mentioned characteristic features. In search for a convincing explanation of the jump phenomenon, we performed detailed numerical simulations of the transient rise of single bubbles in 3D, allowing for a local analysis of the polymer conformation tensor. The latter shows that polymer molecules traveling along the upper bubble hemisphere are stretched in the circumferential direction, due to the flow kinematics. Then, depending on the relaxation time scale of the polymer, the stored elastic energy is either unloaded essentially above or below the bubble's equator. In the former case, this slows down the bubble, while the bubble gets accelerated otherwise. In this latter case, the velocity of motion of the polymer molecules along the bubble is increased, giving rise to a self-amplification of the effect and thus causing the bubble rise velocity to jump to a higher level. Detailed experimental velocity measurements in the liquid field around the bubble confirmed the conclusion that the ratio of the time scale of the Lagrangian transport of polymer molecules along the bubble contour to the relaxation time scale of the polymer molecules determines the sub- or supercritical state of the bubble motion. [Display omitted] • Experiments, 3D DNS and theoretical analysis explain the velocity jump. • The flow around rising bubbles exhibits circumferential polymer stretching. • Release of hoop stress on the downstream bubble part enhances the bubble motion. • The discontinuity is due to a self-amplifying acceleration related to stress relaxation. • The bubble shape and the negative wake result from the same molecular mechanism. [ABSTRACT FROM AUTHOR]