The Errors-in-Variables (EIV) model from statistics is often employed in computer vision though only rarely under this name. In an EIV model, all the measurements are corrupted by noise while the a priori information is captured with a nonlinear constraint among the true (unknown) values of these measurements. To estimate the model parameters and the uncorrupted data, the constraint can be linearized, i.e., embedded in a higher dimensional space. We show that linearization introduces data-dependent (heteroscedastic) noise and propose an iterative procedure, the heteroscedastic EIV (HEIV) estimator to obtain consistent estimates in the most general, multivariate case. Analytical expressions for the covariances of the parameter estimates and corrected data points, a generic method for the enforcement of ancillary constraints arising from the underlying geometry are also given. The HEIV estimator minimizes the first order approximation of the geometric distances between the measurements and the true data points, and thus can be a substitute for the widely used Levenberg-Marquardt based direct solution of the original, nonlinear problem. The HEIV estimator has however the advantage of a weaker dependence on the initial solution and a faster convergence. In comparison to Kanatani's renormalization paradigm (an earlier solution of the same problem) the HEIV estimator has more solid theoretical foundations, which translate into better numerical behavior. We show that the HEIV estimator can provide an accurate solution to most 3D vision estimation tasks, and illustrate its performance through two case studies: calibration and the estimation of the fundamental matrix.