This lecture will give a brief introduction of Hilbert's 17th problem: is every nonnegative polynomial a summation of squares of rational functions? After Artin's affirmative solution, there is a generalization to Positivstellensatz on the solvability of semialgebraic systems. The Positivstellensatz can be applied to solve global optimization problems of polynomial functions. The basic tool is semidefinite programming (SDP), a very nice convex optimization model.