A central problem in Kahler geometry, proposed in 1980s by E. Calabi is to study the existence of a constant scalar curvature metric on a compact Kahler manifold (within a fixed Kahler class). This is a generalization of classical constant Gaussian curvature on surfaces and includes the more famous Kahler-Einstein metrics for special cases (for example, Calabi-Yau manifolds). Since the introduction by Calabi in 1980s, it has been studied extensively and becomes a very fast-growing subject in Kahler geometry. We will summarize briefly Calabi’s problem program, and consider this problem from a point of view of (fourth order) elliptic PDE. We define a notion of weak solution and prove it regularity. This answers in part a conjecture of Xiuxiong Chen. This is joint work with You Zeng from University of Rochester.