Stud. Univ. Babe¸s-Bolyai Math. 61(2016), No. 3, 257–278 Global smoothness preservation and simultaneous approximation by multivariate discrete operators George A. Anastassiou and Merve Kester Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversary Abstract. In this article we study the multivariate generalized discrete singular operators defined on R N , N ≥ 1, regarding their simultaneus global smooth- ness preservation property with respect to Lp norm for 1 ≤ p ≤∞, by using higher order moduli of smoothness. Furthermore, we study their simultaneous approximation properties. Mathematics Subject Classification (2010): 26A15, 26D15, 41A17, 41A25, 41A28, 41A35, 41A80. Keywords: Simultaneous global smoothness, simultaneous approximation with rates, multivariate generalized discrete singular operators, modulus of smooth- ness. 1. Background In [1], Chapter 3, the author defined α [m] j,r :=        (-1) r-j  r j  j -m , if j =1, 2, ..., r, 1 - r ∑ j=1 (-1) r-j  r j  j -m , if j =0, (1.1) for r ∈ N,m ∈ Z + and δ [m] k,r := r X j=1 α [m] j,r j k , k =1, 2, ..., m ∈ N. (1.2) See that r X j=0 α [m] j,r =1. (1.3)