We present direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains. 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 bubbles are small. We perform a detailed study of the rise velocity of a regular array of three-dimensional bubbles at different volume fractions.