What is the maximum number of edges in a graph on n vertices without triangles? Mantel’s answer in 1907—that at most half of the edges can be present—started a new field: extremal combinatorics. More generally, what is the maximum number of edges in an n-vertex graph that does not contain any subgraph isomorphic to H? What about if you consider hypergraphs instead of graphs? I will introduce the technique of sums of squares and discuss how it can be used to attack such problems.