  It is well-known that the Fermat surface of degree $d\geq 3$ has $3d^2$
lines. However, it has not yet been established what is the maximum number of
pairwise disjoint lines that it can have if $d\geq 4$. In this article we show
that the maximal number of skew lines on the Fermat surface of degree $d \geq
4$ is $3d$ either $d$ even or $d$ odd distinct of $5$, otherwise ($d = 5$) it
contains no more than $13$ pairwise disjoint lines.
