Appears in collection : Geometry and algebra of linear matrix inequalities / Géométrie et algèbre des inégalités matricielles linéaires

We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely : - for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form $R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K }$. $D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K }$. by a hierarchy of inner sublevel set approximations $\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right }\subset R_f$. or outer sublevel set approximations $\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right }\supset D_f$. for some polynomiales $(J_k)$ of increasing degree : - for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.