By developing the method introduced by Kobayashi in 1960's, Boyer, Galicki and Kollár found many examples of simply connected Sasaki- Einstein 5-manifolds. For such examples they verified existence of orbifold Kähler-Einstein metrics on various log del Pezzo surfaces, in particular weighted log del Pezzo hypersurfaces. We discuss about recent progresses of the existence problem of Sasaki -Einstein and Kähler-Einstein metric on 5-manifold and weighted del Pezzo hypersurfaces respectively.