In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the etale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic, in the etale topology to the combine, torial data describing affme space endowed with a linear torus action. Finally, we provide an effective method to check the smoothness of a Gm-threefold in terms of the combinatorial data. (C) 2017 Elsevier Inc. All rights reserved.