We discuss algorithms for optimizing polynomials on semialgebraic sets using representation theorems from real algebraic geometry for positive polynomials. In the case of compact semialgebraic sets, the method of Lasserre generates a sequence of SDP relaxations which converge to the solution, however this method does not always work in the non-compact case. We will discuss work of Demmel, Nie, and Sturmfels in the global case and joint work with Demmel and Nie in the case of non-compact semialgebraic sets.