This paper presents algorithms for fitting circles and ellipses to given points in the plane in the least-squares sense, minimizing the sum of squared distances to the points. The authors compare these algorithms with classical methods. Circles and ellipses can be represented algebraically or parametrically. The Gauss-Newton method is used to solve nonlinear least-squares problems. The method involves approximating the solution and solving a linear least-squares problem iteratively. The paper dedicates itself to Åke Björck on the occasion of his 60th birthday. For fitting a circle, the algebraic representation is used: $ F(\mathbf{x}) = a\mathbf{x}^{\mathrm{T}}\mathbf{x} + \mathbf{b}^{\mathrm{T}}\mathbf{x} + c = 0 $. The coefficients a, b, and c are determined by solving a linear system of equations. For m > 3, a non-trivial solution is not expected unless all points lie on a circle. The overdetermined system $ Bu = r $ is solved, minimizing $ \|r\| $. This leads to a standard least-squares problem. The paper also discusses the use of parametric representations for minimizing the sum of squared distances. The authors use the Gauss-Newton method to solve the nonlinear least-squares problem, which involves solving a linear least-squares problem iteratively. The paper is dedicated to Åke Björck, who has contributed significantly to the numerical solution of least-squares problems.This paper presents algorithms for fitting circles and ellipses to given points in the plane in the least-squares sense, minimizing the sum of squared distances to the points. The authors compare these algorithms with classical methods. Circles and ellipses can be represented algebraically or parametrically. The Gauss-Newton method is used to solve nonlinear least-squares problems. The method involves approximating the solution and solving a linear least-squares problem iteratively. The paper dedicates itself to Åke Björck on the occasion of his 60th birthday. For fitting a circle, the algebraic representation is used: $ F(\mathbf{x}) = a\mathbf{x}^{\mathrm{T}}\mathbf{x} + \mathbf{b}^{\mathrm{T}}\mathbf{x} + c = 0 $. The coefficients a, b, and c are determined by solving a linear system of equations. For m > 3, a non-trivial solution is not expected unless all points lie on a circle. The overdetermined system $ Bu = r $ is solved, minimizing $ \|r\| $. This leads to a standard least-squares problem. The paper also discusses the use of parametric representations for minimizing the sum of squared distances. The authors use the Gauss-Newton method to solve the nonlinear least-squares problem, which involves solving a linear least-squares problem iteratively. The paper is dedicated to Åke Björck, who has contributed significantly to the numerical solution of least-squares problems.