1. Introduction: Examples of Inverse Problems.- 1.1. Differentiation as an Inverse Problem.- 1.2. Radon Inversion (X-Ray Tomography).- 1.3. Examples of Inverse Problems in Physics.- 1.4. Inverse Problems in Signal and Image Processing.- 1.5. Inverse Problems in Heat Conduction.- 1.6. Parameter Identification.- 1.7. Inverse Scattering.- 2. Ill-Posed Linear Operator Equations.- 2.1. The Moore-Penrose Generalized Inverse.- 2.2. Compact Linear Operators: Singular Value Expansion.- 2.3. Spectral Theory and Functional Calculus.- 3. Regularization Operators.- 3.1. Definition and Basic Results.- 3.2. Order Optimality.- 3.3. Regularization by Projection.- 4. Continuous Regularization Methods.- 4.1. A-priori Parameter Choice Rules.- 4.2. Saturation and Converse Results.- 4.3. The Discrepancy Principle.- 4.4. Improved A-posteriori Rules.- 4.5. Heuristic Parameter Choice Rules.- 4.6. Mollifier Methods.- 5. Tikhonov Regularization.- 5.1. The Classical Theory.- 5.2. Regularization with Projection.- 5.3. Maximum Entropy Regularization.- 5.4. Convex Constraints.- 6. Iterative Regularization Methods.- 6.1. Landweber Iteration.- 6.2. Accelerated Landweber Methods.- 6.3. The ?-Methods.- 7. The Conjugate Gradient Method.- 7.1. Basic Properties.- 7.2. Stability and Convergence.- 7.3. The Discrepancy Principle.- 7.4. The Number of Iterations.- 8. Regularization With Differential Operators.- 8.1. Weighted Generalized Inverses.- 8.2. Regularization with Seminorms.- 8.3. Examples.- 8.4. Hilbert Scales.- 8.5. Regularization in Hilbert Scales.- 9. Numerical Realization.- 9.1. Derivation of the Discrete Problem.- 9.2. Reduction to Standard Form.- 9.3. Implementation of Tikhonov Regularization.- 9.4. Updating the Regularization Parameter.- 9.5. Implementation of Iterative Methods.- 10. TikhonovRegularization of Nonlinear Problems.- 10.1. Introduction.- 10.2. Convergence Analysis.- 10.3. A-posteriori Parameter Choice Rules.- 10.4. Regularization in Hilbert Scales.- 10.5. Applications.- 10.6. Convergence of Maximum Entropy Regularization.- 11. Iterative Methods for Nonlinear Problems.- 11.1. The Nonlinear Landweber Iteration.- 11.2. Newton Type Methods.- A. Appendix.- A.1. Weighted Polynomial Minimization Problems.- A.2. Orthogonal Polynomials.- A.3. Christoffel Functions.