Classical results of Hilbert, such as the Hilbert basis theorem and Hilbert's syzygy theorem, show that polynomials in a fixed number n of variables are not too complicated. Recent work of a number of people have established a complementary principle: polynomials of a fixed degree d, but in an arbitrary number of variables, are also not too complicated. I will give an overview of this recent work. Speaker(s): Andrew Snowden (University of Michigan)