Running title: Numerical Simulations

An extensive account of the available numerical methodologies for capturing the dynamic behaviour and interaction of translating and/or oscillating bubbles is provided. Different numerical schemes are classified in three large categories according to the way by which they capture deformation of the interface, as: boundary integral methods where only the bubble’s interface is discretized, boundary fitted schemes where the flow field surrounding the bubble is discretized with the interface being identified as one of the coordinate surfaces, and surface tracking schemes where the entire domain both inside and outside the bubble is discretized while the shape of the interface is described via a volume-of-fluid, a level-set or a polynomial function. The boundary integral method provides a better description of the fine details during the last stages of bubble collapse and break-up. The approach via boundary fitted coordinates allows for proper description of the vorticity field around a bubble as well as the formation of eddies behind a translating bubble, over a wide range of Re. Surface tracking techniques can capture severe topological changes in the bubble shape and consequently provide a means to capture the post break-up or post coalescence dynamics. The impact of recent advances in numerical simulations in the understanding of issues of fundamental importance in bubble dynamics, such as monitoring the details of bubble deformation during collapse and identifying the factors determining the mechanism that triggers path instabilities of rising bubbles, is discussed. Recent developments in the area of direct numerical simulations of bubble dynamics are presented. I