A new, broad family of quantile-based estimators is described, and theoretical and empirical evidence is provided for their robustness to outliers in the response. The proposed method can be used to estimate all types of parameters, including location, scale, rate and shape parameters, extremes, regression coefficients and hazard ratios, and can be extended to censored and truncated data. The described estimator can be utilized to construct robust versions of common parametric and semiparametric methods, such as linear (Normal) regression, generalized linear models, and proportional hazards models. A variety of significant results and applications is presented to show the flexibility of the proposed approach. The R package Qest implements the estimator and provides the necessary functions for model building, prediction, and inference.