In 1975, Szemer\'edi proved that any subset of the natural numbers with positive upper density must contain arbitrarily long finite arithmetic progressions. Szemer\'edi's original argument was purely combinatorial, and then Furstenberg gave an alternative proof using ergodic theory a couple of years later. Objects called "nonconventional ergodic averages" appeared for the first time in Furstenberg's proof, and understanding the limiting behavior of very general such averages became an important problem in ergodic theory. After breakthrough work of Bourgain in the late 1980s and early 1990s, no further progress had been made on proving pointwise almost everywhere convergence of these averages until recently. I will report on this progress, along with some of the key inputs from additive combinatorics, focusing on very recent joint work of mine with Dariusz Kosz, Mariusz Mirek, Renhui Wan, and Jim Wright addressing a question of Bergelson from 1996.