Recently, Anton Mellit and I gave a proof of the famous shuffle conjecture of Haglund, Haiman, Loehr, Ulyanov, and Remmel, which predicts a combinatorial formula for the character of the diagonal coinvariant algebra, and other quantities in algebraic geometry. I'll explain what this conjecture is about, and explain the algebraic structures that go into this recent proof. Hopefully if there's time, I'll explain some of remarkable unsolved generalizations, and their role in algebraic geometry.