We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space CPn, where n = 3 and 4. Let M2n be a closed smooth 2n-manifold homotopy equivalent to CPn. We show that, up to diffeomorphism, M6 has a unique differentiable structure and M8 has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N2n of CPn for n = 4,7 or 8 and six distinct differentiable structures on N10. © Indian Academy of Sciences.