The uniformization theorem in one complex variable states that a simply connected Riemann surface is either isomorphic to the Riemann sphere, the Poincare disk, or the complex plane. The uniformization conjecture, proposed by Yau in 1974, looks for possible generalizations in higher dimensions. The conjecture states that a complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to the complex Euclidean space. In this talk, I will discuss some recent progress on this conjecture and related problems.