In this work, we propose some approximations for the conditional quantile of one component of a random vector, given the other components. We focus on the case where the vector has an elliptical distribution. We first give exact expressions for conditional quantiles, and discuss problems that occur for computing these values. A first affine regression quantile estimator is detailed, its distribution is given, and direct simple expressions are derived for some particular elliptical distributions. The performance of this regression quantile is shown to be very poor for extremal quantile levels, so that a second approximation is proposed. We prove that this new extremal approximation is asymp-totically equivalent to the true conditional quantile. Through numerical illustrations, the study shows that for usual techniques as Kriging, Quan-tile Regression may perform poorly when one leaves the usual Gaussian random field assumption, thus justifying the use of proposed extremal quantile approximations.