Problems in extremal graph theory typically aim to maximize some graph parameters under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, is the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from the geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graphs had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore.