Second-order and fourth-order methods for approximate solutions of viscous Hamilton-Jacobi equations are developed on the basis of the weighted essentially non-oscillator (WENO) scheme.By modifying the numerical flux,constructing the second-order approximation based on nonlinear limiter and fourth-order approximation based on Taylor expansion for viscosity term, the one- and two-dimensional viscous Hamilton-Jacobi equations are solved successfully. Numerical tests demonstrate the desired high-resolution,robustness and non-oscillatory behaviors of the schemes.