Abstarct: This thesis consists of two papers on numerical approximation of the Cahn-Hilliard equation. The main part of the work is concerned with the Cahn-Hilliard equation perturbed by noise, also known as the Cahn-Hilliard-Cook equation. In the first paper we consider the linearized Cahn-Hilliard-Cook equation and we discretize it in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The analysis is set in a frxamework baxsed on analytic semigroups.The main part of the work consists of detailed error bounds for the corresponding deterministic equation. In the second paper we study the nonlinear Cahn-Hilliard-Cook equation. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate.