Tractable characterizations of polynomials which are nonnegative on a set K, is a topic of independent interest in Mathematics, but is also of primary importance in many important applications, and notably in global optimization. We will review two kinds of tractable characterizations for nonnegative polynomials through linear matrix inequalities (LMIs) and semidefinite programs (SDPs). The first characterization of nonnegativity is based on the defining polynomials of the set K, via sums of squares (SOS)-weighted representation. For instance, in global optimization this allows to define a hierarchy of semidefinite relaxations which yields a monotone sequence of lower bounds converging to the global optimum (and in fact, finite convergence is generic). The second (dual) characterization of nonnegativity is based on moments of a measure whose support is K. In this approach, checking nonnegativity is reduced to solving a sequence of generalized eigenvalue problems. When applied in global optimization over K, this results in a monotone sequence of upper bounds converging to the global minimum, which complements the previous sequence of lower bounds. These two (dual) characterizations provide convex inner (resp. outer) approximations (by spectrahedra) for the convex cone of polynomials that are nonnegative on K.