Some results of a computational study of axisymmetric Taylor bubbles rising in a vertical tube filled with a quiescent, upward- or downward-flowing liquid are presented. The computational method is based on a finite-volume discretization of the Navier-Stokes equations on a staggered grid. The interface is maintained sharp and is discretized using a set of marker points linked by cubic splines. Independent theoretical results and published data are used for its validation. The simulations illustrate the effect of the Morton and Eotvos numbers on several steady-state and time-dependent processes.