Inverse Problems is an interdisciplinary research area with profound applications in many areas of science, engineering, technology, and medicine. Nowadays, a core technique for solving imaging problems are regularization methods. The foundations of these approximation methods were laid by Tikhonov decades ago, when he generalized the classical definition of well-posedness. In the early days of regularization methods, they were analyzed mostly theoretically, while later on numerics, efficient solutions, and applications of regularization methods became important. This Minitutorial gives a survey on theoretical developments in regularization theory: Starting from quadratic regularization methods for linear ill-posed problems, to convex regularization, and to non-convex regularization methods of non-linear problems. The theoretical analysis will be supported by particular imaging examples.