The errors-in-variables (EIV) regression model, being more realistic by accounting for measurement errors in both the dependent and the independent variables, is widely used in econometrics, chemistry, medical, and environmental sciences, etc. The traditional EIV model estimators, however, can be highly biased by outliers and other departures from the underlying assumptions. In this work, we propose two novel nonparametric estimation approaches - the least sine squares (LSS) and the robust compound regression (RCR) analysis methods for the robust estimation of EIV models. The RCR method, as a natural extension and combination of the new LSS method and the compound regression analysis method developed in our own group (Leng and Zhu 2009), provides the robust counterpart of the entire class of the traditional maximum likelihood estimation (MLE) solutions of the EIV model, in a 1-1 mapping. The advantages of both new approaches lie in their intuitive geometric interpretations, their distribution free property, their independence to the ratio of the error variances, and most importantly their robustness to outlier contamination and other violations of distribution assumptions. Monte Carlo studies are conducted to compare these new robust EIV model estimation methods to other nonparametric regression analysis methods including the least squares (LS) regression analysis method, the orthogonal regression (OR) analysis method, the geometric mean regression (GMR) analysis method, and the robust least median of squares (LMS) regression analysis method. Guidelines on which regression methods are suitable under what circumstances are provided through these simulation studies as well. Real-life examples are provided to further illustrate and motivate these new approaches.