Prerequisites: Linear algebra, multi-variable calculus. The aim of the course is to introduce the basic notions and difficulties encountered with ill-posed inverse problems, to present methods for analyzing these problems and to give some tools that enable to solve such problems. The course will show what a regularization method is and introduce different kinds of regularization techniques and the basic properties of these methods for linear ill-posed problems. Non-linear inverse problems are also studied through some examples: inverse spectral problem, inverse problem of electrical impedance tomography and the inverse scattering problem. The course will introduce numerical tools for analyzing inverse problems, with a focus on the adjoint state method. The Bayesian estimation is also considered. Examples of inverse problems are provided especially in medical imaging.