This article considers the existence and regularity of Kahler Einstein metrics on a compact Kahler manifold M with edge singularities with cone angle 2 pi beta along a smooth divisor D. We prove existence of such metrics with negative, zero and some positive cases for all cone angles 2 pi beta <= 2 pi. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along D for all 2 pi beta < 2 pi.