In this talk, I will introduce the Kakeya problem and the contributions of Bourgain, Wolff and Tao etc. I will focus on a new proof of Wolff 's L^(5/2)-bound for the Kakeya maximal function in three dimensional Euclidean space. We reprove his result without using the argument of induction on scales. This method can be extended to the higher dimensions and cover Wolff 's L^(d+2)/2 estimate. In particular, it can be applied to evaluating the Hausdorff dimension of Nikodym sets in manifolds of constant curvature.