Matrix low-rank approximation is intimately related to data modelling by a linear system; a problem that arises frequently in many different fields. This book is a comprehensive exposition of the theory, algorithms, and applications of structured low-rank approximation. Efficient local optimization methods and effective suboptimal convex relaxations for Toeplitz, Hankel, and Sylvester structured problems are presented. A major part of the text is devoted to application of the theory. Applications described include: • system and control theory: approximate realization, model reduction, output error and errors-in-variables identification; • signal processing: harmonic retrieval, sum-of-damped exponentials, finite impulse response modelling, and array processing; • computer algebra for approximate factorization and common divisor computation; • computer vision for image deblurring and segmentation; • machine learning for information retrieval and clustering; • bioinformatics for microarray data analysis; • chemometrics for multivariate calibration; and • psychometrics for factor analysis. Special knowledge from the respective application fields is not required. The book is complemented by a software implementation of the methods presented, which makes the theory directly applicable in practice. In particular, all numerical examples in the book are included in demonstration files and can be reproduced by the reader. This gives hands-on experience with the theory and methods detailed. In addition, exercises and MATLAB® examples will assist the reader quickly to assimilate the theory on a chapter-by-chapter basis. Data Approximation by Low-complexity Models is a broad survey of the theory and applications of its field which will be of direct interest to researchers in system identification, control and systems theory, numerical linear algebra and optimization. The supplementary electronic lecture slides, problems and solutions render it suitable for use in teaching graduate courses in those subjects as well.