In 2008, Zeev Dvir gave a surprisingly short proof of the Kakeya conjecture over finite fields:  a finite subset of F_q^n containing a line in every direction has cardinality at least c_n q^n.  The "polynomial method" introduced by Dvir has led to a wave of activity in applications of algebraic and arithmetic geometry to extremal problems in combinatorial geometry, including a theme semester at IPAM in spring 2014.  I'll give a general talk about this line of work, including results of Guth and Katz, Kollar, and some of my own (joint with Oberlin-Tao and Hablicsek) and talk about some open questions and ideas for further progress.