Bilodeau and Duchesne [5] extended the S-estimates introduced by Davies [7] for multivariate location and scatter; then Van Aelst and Willems [26] studied the robustness of these estimators. Agull´ o et al. [1] extended the minimum covariance determinant estimate introduced by Rousseeuw [21] and Roelandt et al. [20] extended the definition of GS-estimates introduced by Croux et al. [6]. These estimates have a high breakdown point but are not highly efficient when the errors are Gaussian and q is small. In order to solve this, Agull´ o et al. [1] improved the efficiency of their estimates, maintaining their high breakdown point, by considering one-step reweighting and one-step Newton-Raphson GM-estimates. Garc´ıa Ben et al. [8] extended τ -estimates for multivariate regression, obtaining a estimate with high breakdown point and a high Gaussian efficiency. Another important approach to obtain robust and efficient estimates is contrained M (CM) estimation, proposed by Mendes and Tyler [17] for regression and by Kent and Tyler [13] for multivariate location and scatter. The bias of CM estimates for regression was studied by Berrendero et al. [3]. Following this approach, Bai et al. [2] proposed CM estimates for the multivariate linear model. In this paper we propose robust estimates for the linear model based on the MM approach, first proposed by Yohai [27] for the univariate linear model, and later by Lopuha¨ a [15], Tatsuoka et al. [25] and Salibi´ an-Barrera et al. [23] for multivariate location and scatter. We show that our estimates have both a high breakdown point and a high normal efficiency. In Section 2 we define MM-estimates for the MLM and prove some properties. In Section 3 and 4 we study their breakdown point and Influence Function. In Section 5 and 6 we study the asymptotic properties (consistency and asymptotic normality) of the MM-estimates assuming random predictors and errors with an elliptical unimodal distribution. In Section 7 we describe a computing algorithm 3