Abstract: In this paper, a fourth order WENO (Weighted Essentially Non-Oscillatory) scheme is proposed for nonlinear degenerate parabolic equations. A new fourth order WENO reconstruction method is proposed to discretize the second order spatial derivative term and then a fourth order Runge-Kutta method is employed to advance in time. The linear weights of the proposed scheme can be any positive numbers with the symmetry requirements and that their sum equals one. Besides, the proposed scheme involves no mapping procedure and negative weights. Compared with the sixth order WENO schemes which are based on a six points stencil, the proposed scheme utilizes a four points stencil. Therefore, the proposed scheme is easier to be extended to unstructured meshes and the treatments of the boundary conditions is easier to deal with. Finally, several numerical examples are provided to illustrate the fourth order accuracy, the non-oscillatory property and the efficiency of the proposed scheme.