A method is described for computing the motion of bubbles through a liquid under conditions of large Reynolds and finite Weber numbers. Ellipsoidal harmonics are used to approximate the shapes of the bubbles and the flow induced by the bubbles, and a method of summing flows induced by groups of bubbles, using a fast multipole expansion technique is employed so that the computational cost increases only linearly with the number of bubbles. Several problems involving one, two and many bubbles are examined using the method. In particular, it is shown that two bubbles moving towards each other in an impurity-free, inviscid liquid touch each other in a finite time. Conditions for the bubbles to bounce in the presence of non-hydrodynamic forces and the time for bounce when these conditions are satisfied are determined. The added mass and viscous drag coefficients and aspect ratio of bubbles are determined as a function of bubble volume fraction and Weber number.