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Least squares problems arise when one attempts to fit a model y = n(x,beta) to points (y1,x1),...,(yn,xn). Solutions to such problems are obtained by optimizing the sum of squared deviations over an admissible region. This paper discusses the basic theory of optimization for a general objective function and applies this material to both the linear and nonlinear least squares problems. In linear least squares normal equations for both the full rank and less than full rank cases are considered and the Kuhn-Tucker conditions are used to obtain the normal equations under linear inequality constraints. In nonlinear least squares, different iterative procedures, which may be used to obtain a solution, are discussed. The methods considered are steepest descent, Newton-Raphson, Gauss-Newton, Hartley's modified Gauss-Newton, and that of Marquardt. Results are obtained which relate Marquardt's method to equality constrained least squares. (Author)