  We introduce a direct image formalism for constructible motivic functions.
One deduces a very general version of motivic integration for which a change of
variables theorem is proved. These constructions are generalized to the
relative framework, in which we develop a relative version of motivic
integration. These results have been announced in math.AG/0403349 and
math.AG/0403350.
  Main results and statements unchanged. Many minor slips corrected and some
details added.
