This paper deals with the scattering and diffraction of a plane wave by a randomly rough half-plane by three tools: the small perturbation method, the Wiener-Hopf technique and a group theoretic consideration based on the shift-invariance of a homogeneous random surface. For a slightly rough case, the scattered wavefield is obtained up to the second-order perturbation with respect to the small roughness parameter and represented by a sum of the Fresnel integrals with complex arguments, integrals along the steepest descent path and branch-cut integrals, which are evaluated numerically. For a Gaussian roughness spectrum, intensities of the coherent and incoherent waves are calculated in the region near the edge and illustrated in figures, in terms of which several characteristics of scattering and diffraction are discussed.