We present a number of three-dimensional axisymmetric free boundary pr
oblems for two immiscible fluids, such as air and water. A level set m
ethod is used where the interface is the zero level set of a continuou
s function while the two fluids are solutions of the incompressible Na
vier-Stokes equation. We examine the rise and distortion of an initial
ly spherical bubble into cap bubbles and toroidal bubbles. Steady solu
tions for gas bubbles rising in a liquid are computed, with favourable
comparisons to experimental data. We also study the inviscid limit an
d compare our results with a boundary integral method. The problems of
an air bubble bursting at a free surface and a liquid drop hitting a
free surface are also computed.