This project deals with approximation of solutions for systems of ordinary first-order differential equations. Peano's existence theorem and Osgood's uniqeness theorem will be proved. The following methods; Euler's method, the Improved Euler method and Heun's method, will be explained and supported by underlying theory. A short general introduction to methods based on Taylor expansion will be made. Furthermore Runge-Kutta methods will be introduced, among these the classical Runge-Kutta formula of order 4, RK4. Numerical experiments concerning Euler's method, the Improved Euler method, Heun's method and RK4, will be carried out, to illustrate the underlying theory. Throughout the project the use and effectiveness of the mentioned methods will be illustrated by small examples, in which the general solution is known. Finally an example will be carried out, in which no general solution exists. This example shows the effects a topspin will have on the trajectory of a tennisball.