[en] Recent developments in the theoretical treatment of Compton scattering from bound electrons will be examined. The utility of simple approximate methods in calculating single ionization have recently been investigated using a relativistic S matrix computer code. The region of validity of the A² term of the nonrelativistic photon electron interaction has been determined. The extension of this approximation to the calculation of double ionization will be discussed. Particular emphasis will be placed on the double ionization of helium. However, some consideration will also be given to the double ionization of other atoms