Let k[X] be a polynomial ring in n≥3 variables over a field k of characteristic zero.We study constants and Darboux polynomials of some class of derivations of k[X].We present several new examples of homogeneous derivations of k[X] without Darboux polynomials.We present some general properties of monomial derivations of k[X],and we characterize some large class of monomial derivations without Darboux polynomials.In particular,we describe all monomial deriva-tions of k[x; y; z] which have no Darboux polynomials.Moreover,we describe polynomial constants and rational constants of cyclotomic derivations of k[X].