We consider the problem of vanishing of the moments View the MathML sourcemk(P,q)=∫ΩPk(x)q(x)dμ(x)=0,k=1,2,…, Turn MathJax on with Ω a compact domain in RnRn and P(x)P(x), q(x)q(x) complex polynomials in x∈Ωx∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture: Conjecture A. For positive μ if mk(P,1)=0mk(P,1)=0for k=1,2,…k=1,2,… , then mk(P,q)=0mk(P,q)=0for k≫1k≫1for any q. We recall recent results on one-dimensional (MVP) obtained in Muzychuk and Pakovich (2009) [24], Pakovich (2009,2004) [25] and [26], Pakovich (preprint) [28] and prove some initial results in several variables, stressing the role of the positivity assumption on the measure μ. On this base we analyze some special cases of Conjecture A and provide in these cases a complete characterization of the measures μ for which this conjecture holds.