For a normal toric projective variety \(X\) over \({\mathbb C}\), by [D. A. Cox, J. Algebr. Geom. 4, No. 1, 17–50 (1995; Zbl 0846.14032)] there exists a GIT quotient \(p:\hat{X}\rightarrow X\) given by the action of a quasitorus, that is, \(\hat{X}\) is the complement in \(\text{Spec}\, R(X)\) of the closed set defined by the irrelevant ideal \(J\) of the Cox ring \(R(X)\) graded by the divisor class group \(\text{Cl}(X)\). Here, the quasitorus \(G=\text{Spec}\,{\mathbb C}[\text{Cl}(X)]\) acts on \(R(X)\) and \(\hat{X}\) by the \(\text{Cl}(X)\)-grading. Using this description of \(X\), a hypersurface in \(X\) can be defined as the image under \(p\) of the zero set of a homogeneous element \(f\in R(X)\) and, following [V. I. Danilov, Lect. Notes Math. 1479, 26–38 (1991; Zbl 0773.14011)] and [V. V. Batyrev and D. A. Cox, Duke Math. J. 75, No. 2, 293–338 (1994; Zbl 0851.14021)], such hypersurface is called quasi-smooth if the zero set \(V(f)\) is smooth in \(\hat{X}\), that is, the singular locus of \(f\) is contained in the irrelevant locus. The main result of the paper under review gives combinatorial conditions for the quasi-smoothness of a general member \(Y\) of a monomial linear system \({\mathcal L}\) on \(X\), first in terms of its Newton polytope in Theorem 3.6 and Corollary 3.8, and next in terms of the matrix of exponents of a monomial basis defining \(Y\) in homogeneous coordinates in Theorem 4.1. As applications, in Section 5 of the paper the authors recover the known classification of quasi-smooth hypersurfaces in fake weighted projective spaces, and in Section 6 they use quasi-smoothness to define some families of Calabi-Yau hypersurfaces.