  In this paper we give a classification of cohomogeneity one manifolds
admitting an invariant metric with quasipositive sectional curvature except for
two $7$-dimensional families. The main result carries over almost verbatim from
the classification results in positive curvature carried out by Verdiani and
Grove, Wilking and Ziller. Three main tools used in the positively curved case
that we generalized to quasipositively curved cohomogeneity one manifolds are
Wilking's Chain Theorem, the classification of positively curved fixed point
homogeneous manifolds by Grove and Searle and the Rank Lemma.
