Manifolds with non-negative sectional curvature are of interest since the beginning of the global Riemmannian geometry. Though there are many examples, the methods of construction are few. K.Grove and W.Ziller have generalized a gluing method of J.Cheeger to the cohomogeneity one manifolds and showed a rich class of non-negatively curved manifolds. However not all cohomogeneity one manifolds support an invariant non-negatively curved metric. K.Grove, L.Verdiani, B.Wilking and W.Ziller discovered the first examples of the obstruction. I will show more examples of the obstruction. Also I will report the current state of the classification of the cohomogeneity one manifold who has a small family of the invariant metrics.