Ill-posed inverse problems have wide applications in many fields such as oceanography, signal processing, machine learning, biomedical imaging, remote sensing, geophysics, and others. In this dissertation, we address the problem of solving unstable operator equations with iteratively regularized Newton-type algorithms. Important practical questions such as selection of regularization parameters, construction of generating (filtering) functions based on a priori information available for different models, algorithms for stopping rules and error estimates are investigated with equal attention given to theoretical study and numerical experiments.