  Oort has conjectured that there do not exist Shimura curves lying generically
in the Torelli locus of curves of genus $g \geq 8$. We show that there do not
exist one-dimensional Shimura families of semi-stable curves of genus $g\geq 5$
of Mumford type. We also show that there do not exist Shimura curves lying
generically in the Torelli locus of hyperelliptic curves of genus $g\geq 8$.
The first result proves a slightly weaker form of the conjecture for the case
of Shimura curves of Mumford type. The second result proves the conjecture for
the Torelli locus of hyperelliptic curves. We also present examples of Shimura
curves contained generically in the Torelli locus of curves of genus $3$ and
$4$.
