In this thesis, numerical methods for a certain class of singular initial value problems of the second order are investigated. The second order problems are transformed to first order systems with a singularity of the first kind and solved using one-step methods (implicit and explicit Euler method, trapezoidal rule and box scheme). It is proven that the implicit and explicit Euler method and the trapezoidal rule show their classical convergence properties (order 1 and 2, respectively), whereas for the box scheme an order reduction occurs which can also be observed in experiments. Finally, an asymptotic error expansion for the implicit Euler method is derived, which is frequently a basis for the use of certain acceleration techniques. The solvability of a certain class of singular nonlinear initial value problems is discussed. Particular attention is paid to the structure of initial conditions necessary for a bounded solution to exist. The implicit Euler rule applied to approximate the solution of the singular system is shown to be stable and to retain its classical convergence order. Moreover, the asymptotic error expansion for the global error of the above approximation is proven to have the classical structure. Finally, experimental results showing the feasibility of the approximation obtained by the Euler method to serve as a basic method for the acceleration technique known as the Iterated Defect Correction are presented.