If a real polynomial $f$ can be written as a sum of squares of real polynomials, then clearly $f$ is nonnegative on $\mathbb {R} ^n$, and an explicit expression of $f$ as a sum of squares is a certificate of positivity for $f$. This idea, and generalizations of it, underlie a large body of theoretical and computational results concerning positive polynomials and sums of squares. In this survey article, we review the history of the subject and give an overview of recent results, both theoretical results concerning the existence of certificates of positivity and work on computational and algorithmic issues.