Many problems in discrete geometry can be naturally encoded in a structure known as a semialgebraic graph. These include the Erdős unit distance problem, questions about incidences of algebraic objects, and more. I will discuss several new structural and extremal results about semialgebraic graphs. These include a very strong regularity lemma with optimal bounds and improvements to the Zarankiewicz problem and the Erdős–Hajnal problem for semialgebraic graphs. These results are proved via a novel extension of the polynomial method, building on the polynomial partitioning machinery of Guth–Katz and Walsh. Based on joint work with Hung-Hsun Hans Yu.