Standard least squares algorithms for finding the best-fit geometric surface to coordinate data implicitly assume that the uncertainties associated with the coordinates are uncorrelated and axis-isotropic, i.e., the uncertainties associated with the x-, y- and z-coordinates are equal (but can vary from point to point). Very few coordinate measuring systems have such uncertainty characteristics but, in the absence of quantitative information about the true uncertainty structure, these assumptions can be justified. However, more effort is now being applied to evaluate the uncertainties associated with coordinate measuring systems and the question arises of how best to use this extra information, for example in surface fitting. This paper describes algorithms for fitting geometric surfaces to coordinate data with general uncertainty structure and shows how these algorithms can be implemented efficiently for a class of uncertainty matrices that arise in many practical systems. The fitting algorithms are illustrated on data simulating laser tracker and coordinate measuring machine measurements.