The purpose of this paper is to obtain some direct results for the Durrmeyer variant of q-Bernstein-Schurer operators for functions of one variable introduced by Acu et al. . We also propose to study the bivariate extension of these operators and discuss the rate of convergence by using the modulus of continuity, the degree of approximation for the Lipschitz class of functions and the Voronovskaja type asymptotic theorem. Furthermore, we show the convergence of the operators by illustrative graphics in Maple to certain functions in both one and two dimensional cases. (C) 2015 Elsevier Inc. All rights reserved.