a multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{r}^n$. obvious examples of non-negative polynomials are squares and sums of squares. what is the relationship between non-negative polynomials and sums of squares? i will review the history of this question, beginning with hilbert's groundbreaking paper and hilbert's 17th problem. i will discuss why this question is still relevant today, for computational reasons, among others. i will then discuss my own research which looks at this problem from the point of view of convex geometry. i will show how to prove that there exist non-negative polynomials that are not sums of squares via ``naive" dimension counting. i will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares.