We prove that for a generic element in a nonhyperelliptic component of an abelian stratum Hg(μ) in genus g, the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition μ has positive entries. As a consequence we deduce that all nonhyperelliptic components of H9(μ) are uniruled when μ is a positive partition of 16 and all nonhyperelliptic components of H 2 g (μ) are uniruled when μ is a positive partition of 4g − 4 and either 3 g 5 or g = 6 and l(μ) 4.