  The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$
of a closed manifold is isotopic to a map realizing the Nielsen number of $f$,
which is a lower bound for the number of fixed points among all maps homotopic
to $f$. The main theorem of this paper proves this conjecture for all
orientation preserving maps on geometric or Haken 3-manifolds. It will also be
shown that on many manifolds all maps are isotopic to fixed point free maps.
  The proof is based on the understanding of homeomorphisms on 2-orbifolds and
3-manifolds. Thurston's classification of surface homeomorphisms will be
generalized to 2-dimensional orbifolds, which is used to study fiber preserving
maps of Seifert fiber spaces. Maps on most Seifert fiber spaces are indeed
isotopic to fiber preserving maps, with the exception of four manifolds and
orientation reversing maps on lens spaces or $S^3$. It will also be determined
exactly which manifolds have a unique Seifert fibration up to isotopy. These
informations will be used to deform a map to certain standard map on each piece
of the JSJ decomposition, as well as on the neighborhood of the decomposition
tori, which will make it possible to shrink each fixed point class to a single
point, and remove inessential fixed point classes.
