This document presents the results of a Monte Carlo study of the most commoly discussed methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares. The methods we examine are the three variants of the linearization method, the likelihood method, and the lack-of-fit method. the linearization method is the most frequently implemented method. It is computationally inexpensive and produces easily understandable results. The likelihood and lack-of-fit methods both are much more expensive and more difficult to report. Based on our results, we conclude that among the three variants of the linearization method, the variant based solely on the Jacobina appears perferable because it is simpler, less expensive, more numerically stable, and at least as accurate as the other two variants which utilize the full Hessian. In our tests, however, all three variant of the linearization method often produce gross underestimates of confidence regions and sometimes produce significant underestimates of confidence intervals. Both the likelihood and lack-of-fit methods, on the other hand, perform very reliably. For datasets analyzed, the Bates and Watts curvature measures reliably predict when the linearization method confidence regions will be poor, and for the most part are consistent with our results for the likelihood method.