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We consider moment-like expressions of the form mk=∫ΩPk(x)q(x)dx,whereQ Ω∈Rn is a bounded domain,and P and q are polynomials in x∈ Rn.We study the "Moment vanishing problem" for mk,i.e.our goal is to give necessary and sufficient conditions on Ω,P,q for vanishing of mk,k= Recently,moment vanishing problems of various specific forms appeared in Qualitative Theory of ODEs,in Inverse problems(in particular,in Algebraic signal sampling),in Representation Theory,and in study of Algebras of Differential Operators,as related to the Jacobian Conjecture.In particular,let us consider the Abel differential equation y= p(x)y2 + q(X)y3.(1) A solution y(x)of(1)is called "closed" on [a,b] if y(a)=y(b).Study of closed solutions of(1)is directly related to the classical Hilbert 16-th(= Smale 13-th) and the Poincare Center-Focus problems.The last problem consists(in case of equation(1))in providing necessary and sufficient conditions on p,q,a,b for (1)to have all its solutions closed(center).It turns out that a rather accurate "first order" approximation of such "center conditions" is provided by the vanishing of mk = ∫ab Pk(x)q(x)dx,where P= ∫p.Higher order approximations of the center conditions are provided by the vanishing of the higher Melnikov functions,which are linear combinations of iterated integrals of the form ∫p∫q∫q … ∫p.In recent 20 years an important algebraic-analytic structure has been connected to the Moment vanishing: Composition Algebra of polynomials.The study of this structure significantly improved our understanding ofmoment vanishing,as well as of closed solutions of(1).Recently,a complete and effective solution of the one-dimensional polynomial Moment vanishing problem has been finally produced by F.Pakovich.On this base,a serious progress has been achieved in study of center conditions for polynomial Abel equation(1).In this talk I plan to present some of these new results and some open questions related to moment vanishing.In particular,I plan to present in some detail recent results of Pakovich,stressing the role of the solution spaces of the Moment vanishing problem(which I would like to call "Pakovich spaces").I plan also to outline very recent results of M.Briskin on vanishing of iterated integrals and higher Melnikov functions on Pakovich spaces.