 Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients of representation of Lie groups "G".This description provides a uniform framework for proving many hitherto disparate properties of special functions, such as addition formulas, certain recurrence relations, orthogonality relations, integral representations, and eigenvalue properties with respect to differential operators.Special functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generalizations, Legendre and Jacobi orthogonal polynomials and Bessel functions all arise as matrix coefficients of representations of Lie groups.