This thesis deals with two iterative methods that are used to solve nonlinear ill-posed problems, the Landweber iteration, and the iteratively regularized Gauss-Newton method. Both methods are introduced theoretically and the main results on convergence and convergence rates that were established in recent times are reviewed. In order to prove these results, in either case, very similar assumptions are required which makes it easier to compare them. The second part of the work is concerned with the numerical implementation and comparison of results for different integral operators of the Hammerstein type. The focus therein is on showing that the necessary assumptions are fulfilled and that the numerical tests agree with the theoretical results, as well as comparing convergence rates and CPU time between the Landweber iteration and the iteratively regularized Gauss-Newton method. We will, however, also see cases in which the theoretical assumptions that are required for the convergence rates are not fulfilled or cannot be shown, but where an appropriate solution and even rates can be achieved by the same algorithms.