  This survey article discusses some results on the structure of families
f:V-->U of n-dimensional manifolds over quasi-projective curves U, with
semistable reduction over a compactification Y of U. We improve the Arakelov
inequality for the direct images of powers of the dualizing sheaf. For families
of Abelian varieties we recall the characterization of Shimura curves by
Arakelov equalities. For families of curves we recall the characterization of
Teichmueller curves in terms of the existence of certain sub variation of Hodge
structures. We sketch the proof that the moduli scheme of curves of genus g>1
can not contain compact Shimura curves, and that it only contains a non-compact
Shimura curve for g=3.
