Inverse problems are concerned with the recovery of some unknown quantities involved in a system from the knowledge of specific measurements. Typical examples are:the boundary distance rigidity problem where one would like to recover the metric tensor of a compact Riemannian manifold with boundary from the knowledge of the geodesic distance between boundary points,spectral inverse problems where one tries to recover a compact Riemannian manifold from the scattering matrix or from the Dirichlet data of eigenfunctions,Calderón's inverse conductivity problem where one is interested in recovering the coefficient of a partial differential equation from Cauchy data,and inverse tomography where one is concerned with inverting integral transformation such as the X-ray or the Radon transforms. This History and Overview of thinking is quite natural in engineering and physical sciences where one aims to determine physical quantities from experimental measurements. It is therefore not surprising that the field of Inverse Problems bears a lot of applications to other scientific domains, for instance medical imaging, seismography, oil prospection, radar imaging, etc. Besides, it involves a wide spectrum of mathematical fields, such as harmonic analysis, partial differential equations (PDEs), microlocal analysis, Riemannian geometry, spectral theory, probability etc. up to numerical implementations on the more applied side.