In this thesis, Brent's method and the Brent-Gay version of Brown's method for solving systems of nonlinear equations are studied. A comparison is then made between these two methods, Powell's hybrid method and Brown's original method. The numerical results show that the Brent-Gay version is an improvement of the method and that in the average Brent's method is the best. Finally, listings of the two studied algorithms and of the test functions are given in the appendices.