The effects of co-current flows on a rising Taylor bubble are systematically investigated by a front tracking method coupled with a finite difference scheme based on a projection approach. Both the upward (the co-current flows the same direction as the buoyancy force) and the downward (the co-current moves in the opposite direction of the buoyancy force) co-currents are examined. It is found that the upward co-current tends to elongate the bubble, while the downward co-current makes the bubble fatter and shorter. For large N(f) (the inverse viscosity number), the upward co-current also elongates the skirted tail and makes the tail oscillate, while the downward co-current shortens the tail and even changes a dimpled bottom to a round shape. The upward co-current promotes the separation at the tail, while the downward co-current suppresses the separation. The terminal velocity of the Taylor bubble rising in a moving flow is a linear combination of the mean velocity (U(C)) of the co-current and the terminal velocity (U(0)) of the bubble rising in the stagnant liquid, and the constant is around 2 which agrees with the literature. The wake length is linearly proportional to the velocity ratio (U(C)/U(O)). The co-currents affect the distribution of the wall shear stresses near the bubble, but not the maximum. (C) 2011 Elsevier Ltd. All rights reserved.