  We give a complete classification of local and global conformal biharmonic
maps between any two space forms by proving that a conformal map between two
space forms is proper biharmonic if and only if the dimension is 4, the domain
is flat, and it is a restriction of a M\"obius transformation.
  We also show that proper k-polyharmonic conformal maps between Euclidean
spaces exist if and only if the dimension is 2k and they are precisely the
restrictions of M\"obius transformations. This provides infinitely many simple
examples of proper k-polyharmonic maps with nice geometric structure.
