Abstract: A real polynomial is called nonnegative if it takes only nonnegative values. Sums of squares of polynomials (and rational functions) are obviously nonnegative. I will review the rich history of this area of real algebraic geometry, and the recent connections with optimization. I will explain how this topic is inextricably linked to classical topics in algebraic geometry and commutative algebra, and discuss a specific example of square-free monomial ideals, which connects this area to positive semidefinite matrix completion problems.