This thesis is concerned with the numerical construction of simply closed invariant curves of maps defined on the plane. We develop and discuss a Newton-Raphson method that is based on solving a linear functional equation. By using formal power series analytic solutions are derived and conditions for the existence of a unique 2π-periodic continuous solution are established. In order to approximate this particular solution a basis of functions is introduced and an infinite system of linear equations for the coefficients of the basis is considered. We solve a sequence of finite subsystems with increasing dimension. By using B-splines and Fourier series an algorithm for approximating the invariant curve is derived. The algorithm is tested with explicitly given maps, followed by the application to the Van-der-Pol equation and the logistic map. The implementation is checked extensively and the efficiency of the method is illustrated.