Abstract : Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics [1,3,4]. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Their limit distribution is established when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their order with the oscillations of the quantile function and a von-Mises property of the conditional distribution . A simulation experiment and an illustration on American electric data are presented.  Berlinet, A., Gannoun, A. and Matzner-Lober, E. (2001) Asymptotic normality of convergent estimates of conditional quantiles, Statistics, 35, 139--169.  Daouia, A., Gardes, L. and Girard, S. (2013) On kernel smoothing for extremal quantile regression, Bernoulli, to appear.  Samanta, T. (1989) Non-parametric estimation of conditional quantiles, Statistics and Probability Letters, 7, 407--412.  Stute, W. (1986) Conditional empirical processes, The Annals of Statistics, 14, 638--647.