In this book, the authors present a number of examples which lead to ill-posed problems arising with the processing and interpretation of data of physical measurements. Basic postulates and some results in the general theory of ill-posed problems follow. The exposition also includes problems of analytic continuation from continua and discrete sets, analogous problems of continuation of solutions of elliptic and parabolic equations, the main ill-posed boundary value problem for partial differential equations, and results on the theory of Volterra equations of the first kind. A very broad presentation is given of modern results on the problem of uniqueness in integral geometry and on inverse problems for partial differential equations.