  We address the question concerning the birational geometry of the strata of
holomorphic and quadratic differentials. We show strata of holomorphic and
quadratic differentials to be uniruled in small genus by constructing rational
curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to
genus $3\leq g\leq6$, we construct projective bundles over a rational varieties
that dominate the holomorphic strata with length at most $g-1$, hence showing
in addition that these strata are unirational.
