A numerical method for simulating the motion and deformation of an axisymmetric bubble or drop rising or falling in another infinite and initially stationary fluid is developed based on the volume of fluid(VOF)method in the frame of two incompressible and immiscible viscous fluids under the action of gravity,taking into consideration of surface tension effects.A comparison of the numerical results by this method with those by other works indicates the validity of the method.In the frame of inviscid and incompressible fluids without taking into consideration of surface tension effects,the mechanisms of the generation of the liquid jet and the transition from spherical shape to toroidal shape during the bubble or drop deformation,the increase of the ring diameter of the toroidal bubble or drop and the decrease of its cross-section area during its motion,and the effects of the density ratio of the two fluids on the deformation of the bubble or drop are analysed both theoretically and numerically.