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In 1814 Carl Friedrich Gauß (1777–1855) developed his famous method of numerical integration which dramatically improves the earlier method of Isaac Newton (1643–1727) from 1676. Beside the some historical details in this survey, a formulation of this classical theory in modern terminology using theory of orthogonlity on real line, as well as the characterization, existence and uniqueness of these formulas, are presented. A special attention is devoted to the algorithms for constructing such quadrature formulas for nonclassical weight functions, their numerical stability and the corresponding software. Finally, some recent progress in this subject, as well as new important applications of these methods in several different directions (distributions in statistics and physics, summation of slowly convergent series, etc.) are presented.

Mathematics Subject Classification (2010): 33C47, 42C05, 41A55, 65D30, 65D32.

Keywords: Orthogonal polynomials, moments, recursion coefficients, Gaussian quadrature, weight function.