Geometric fitting — parameter estimation for data subject to implicit parametric constraints — is a very common sub-problem in computer vision, used for curve, surface and 3D model fitting, matching constraint estimation and 3D reconstruction under constraints. Although many algorithms exist for specific cases, the general problem is by no means ‘solved' and has recently become a subject of considerable debate among researchers in statistical vision. This paper describes a new, more direct approach to geometric fitting, formulating it as the explicit recovery of a coherent, statistically optimal set of estimates of the “underlying data points” that gave rise to the observations, together with the estimated constraints which these points exactly verify. The method is implemented using an efficient constrained numerical optimization technique, and is capable of handling large problems with complex, constrained parametrizations. As examples of such problems, we consider the optimal estimation of the fundamental and essential matrices and the trifocal tensor, subject to their full sets of algebraic constraints. We also describe how our approach ‘reduces' to existing geometric fitting methods like gradient-weighted orthogonal least squares, and give a novel approach to robustness based on it.