This talk compares local and global optimality conditions for multivariate polynomial optimization problems. First, we prove that the constraint qualification, strict complementarity and second order sufficiency conditions are all satisfied at each local minimizer, for generic cases. Second, we prove that if such optimality conditions hold at each global minimizer, then a global optimality certificate must be satisfied. Third, we show that Lasserre's hierarchy almost always has finite convergence in solving polynomial optimization under the archimedeanness.