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The structure of the Cesàro function spaces Ces p on both [0, 1] and [0, ∞) for 1 < p ∞ is investigated. We find their dual spaces, which equivalent norms have different description on [0, 1] and [0, ∞). The spaces Ces p for 1 < p < ∞ are not reflexive but strictly convex. They are not isomorphic to any L q space with 1 q ∞. They have "near zero" complemented subspaces isomorphic to l p and "in the middle" contain an asymptotically isometric copy of l 1 and also a copy of L 1 [0, 1]. They do not have Dunford-Pettis property but they do have the weak Banach-Saks property. Cesàro function spaces on [0, 1] and [0, ∞) are isomorphic for 1 < p ∞. Moreover, we give characterizations in terms of p and q when Ces p [0, 1] contains an isomorphic copy of l q .

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