Solving inverse and ill-posed problems is a common yet one of the most challenging tasks faced by a physicist. In this Chapter we discuss the basic properties of linear and non-linear operators involved in solving direct (“forward”) problems and their inverse (“backward”) counterparts, the reasons for their ill-posedness, and measures that can be taken to control and stabilize the inversion procedure by regularization. In the three core Sections we discuss topics that are presumably of greatest interest to physicists: the solution of Fredholm and Volterra integral equations of the first and second kind, the solution of inverse Sturm-Liouville problems, i. e. the reconstruction of the operators from their spectra and known symmetry properties, and the solution of retrospective and source/coefficient-recovery problems for partial differential equations. We present methods of phase retrieval based on the data providing only the intensities. Examples and Problems include stable deconvolution of noisy signals, image deblurring techniques, the inverse Radon transformation, and identifying the shape and refraction index of inhomogeneous domains by inverting the far-field pattern of scattered plane waves.