This is the first book in which noisy information is studied in the context of computational complexity, in other words it deals with the computational complexity of mathematical problems for which available information is partial, noisy and priced. The author develops a general theory of computational complexity of continuous problems with noisy information and gives a number of applications; deterministic as well as stochastic noise is considered. He presents optimal algorithms, optimal information, and complexity bounds in different settings: worst case, average case, mixed worst-average and average-worst, and asymptotic. Particular topics include: existence of optimal linear (affine) algorithms, optimality properties of smoothing spline, regularisation and least squares algorithms (with the optimal choice of the smoothing and regularisation parameters), adaption versus nonadaption, relations between different settings. The book integrates the work of researchers over the last decade in such areas as computational complexity, approximation theory and statistics, and includes many new results. Nearly two hundred exercises are supplied with a view to increasing the reader's understanding of the subject. The material is organised in such a way that it can be used either as a textbook for advanced courses, or as a standard reference for professional computer scientists, statisticians, applied mathematicians, engineers, control theorists and economists.