We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale is a nuisance parameter. First, we show how to convert an ordinary robust estimate for the usual model with symmetric errors to an estimate for the more general model with asymmetric errors. Then, in order to improve efficiency, we introduce the truncated maximum likelihood or TML-estimator. A TML-estimate is computed in three steps: first, an initial high breakdown point estimate is computed; then, observations that are unlikely under the estimated model are rejected; finally, the maximum likelihood estimate is computed with the retained observations. The rejection rule used in the second step is based on a cut-off parameter that can be tuned to attain the desired efficiency while maintaining the breakdown point of the initial estimator (e.g., 50%). Optionally, one can use a new adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Under the model, the influence function of this adaptive TML-estimator (or ATML-estimator) coincides with the influence function of the maximum likelihood estimator. The ATML-estimator is, therefore, fully efficient at the model; nevertheless, its breakdown point is not smaller than the breakdown point of the initial estimator. We evaluate the TML- and ATML-estimators for finite sample sizes with the help of simulations and discuss an example with real data. [authors]