A new variant of integral boundary-layer theory is applied to the flow around axisymmetric fluid bodies (bubbles and drops) at Reynolds numbers of order 100. A fourth order polynomial velocity profile is matched to the inviscid flow solution at some distance from the surface, creating the inflection point and maximum velocity which characterize the actual velocity profile. The boundary layer is defined as the region below the point of maximum velocity. The effects of longitudinal curvature of the body on the form of the boundary-layer equations are discussed, but not used in the final model formulation. Predictions of velocity profile shape, pressure variation, boundary-layer thickness and flow separation point agree well with known results for flow around solid and fluid spheres. Solid-sphere results for Re > 50 have an instability just ahead of the flow separation point, that corresponds to the observed onset of oscillations in the wake at this Re.