Five methods are discussed to fit circles to data. Two of the methods are shown to be highly sensitive to measurement error. The other three are shown to be quite stable in this regard. Of the stable methods, two have the advantage of having closed form solutions. A positive aspect of all of these models is that they are coordinate free in the sense that the same estimating circles are produced no matter where the axes of the coordinate system are located nor how they are oriented. A natural extension to fitting spheres to points in 3-space is also given.