In the rst part of the talk we briey describe the Moment-SOS hierarchy [1, 2], a methodology to solve the

Generalized Moment Problem(GMP) with algebraic data, whose list of potential applications is almost endless,

and global optimization being its simplest instance. In a second part we briey consider the inverse problem

of recovering the algebraic boundary of a basic semi-algebraic set from the sole knowledge of moments of the

Lebesgue measure on the set [4]. Finally, the third part of the talk is devoted to the Christoel function [3], a

well-known tool in theory of approximation and orthogonal polynomials. We will describe how it nicely connects

with the rst two parts of the talk, in particular for recovering the graph of a function from moments of the

measure supported on the graph, but also for its role in a key aspect of algorithmic polynomial optimization.