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The algebraic fitting of circles was first proposed by Delogne (1972) and Kasa (1976). We extend the work by Delogne (1972) and Kasa (1976) and propose fast and almost unbiased weighted least squares to best fit circles. The simulations have shown that two non-iterative versions of the bias-corrected weighted LS method are fast and almost unbiased and perform much better than the naive weighted LS method (without applying any bias correction), the ordinary LS-based methods and the gradient-based weighted LS method in terms of both biases and mean squared errors, no matter whether fitting of circles is strongly or weakly constrained geometrically. Nevertheless, a weak geometrical constraint results in a poor fitting of circles. We accordingly propose the regularized variants of the bias-corrected weighted LS method to fit circles from data with a weak geometrical constraint. The simulations have also shown that the two non-iterative regularized variants fit circles satisfactorily well and consistently perform the best among all the regularization methods under study for ill-conditioned problems of fitting circles. • Removing the bias of computed data to construct almost unbiased estimates for algebraic fitting. • Improving the standard algebraic fitting of circles by Delogne (1972) and Kasa (1976). • Proving that the gradient-based weighting is not statistically optimal. • Proposing the statistically rigorous weighted LS method to estimate the circular parameters. • Proposing a fast, bias-corrected regularization for algebraic fitting of circles. [ABSTRACT FROM AUTHOR]