One gives a survey of the fundamental directions and results up to 1981, inclusively, regarding the topic mentioned in the title.In the last 25 years, under the direct influence of imperative practical needs of various areas of science and engineering, a new branch of mathematics, the theory and methods of solution of ill-posed problems, has been developed in a very intensive manner. Presently one totals close to 3000 publications in this topic. Many interesting and important results obtained up to 1972 are reflected in V. A. Morozov's extensive survey [264]. Since then, the theory of ill-posed problems has been significantly enriched and has become an independent discipline; this is attested by the apparition of numerous monographs devoted entirely or in a large part to this circle of questions ( ) and also by the inclusion of corresponding chapters in university textbooks and manuals [18, 31, 59, 79, 83, 176, 177, 187, 191, 201, 240, etc