Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees relatively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kissing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three-dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed.