Abstract: Given a point and a rational curve in the plane, their bisector curve is rational [Farouki and Johnston 1994a]. However, in general, the bisector of two rational curves in the plane is not rational [Farouki and Johnstone 1994b]. Given a point and a rational space curve, this art icle shows that the bisector surface is a rational ruled surface. Moreover, given two rational space curves, we show that the bisector surface is rational (except for the degenerate case in which the two curves are coplanar).