  Teichmueller curves are geodesic discs in Teichmueller space that project to
an algebraic curve in the moduli space $M_g$. We show that for all $g \geq 2$
Teichmueller curves map to the locus of real multiplication in the moduli space
of abelian varieties. Remark that McMullen has shown that precisely for $g=2$
the locus of real multiplication is stable under the $\SL_2(\RR)$-action on the
tautological bundle $\Omega\M_g$.
  We also show that Teichmueller curves are defined over number fields and we
provide a completely algebraic description of Teichmueller curves in terms of
Higgs bundles. As a consequence we show that the absolute Galois group acts on
the set of Teichmueller curves.
