Consider a system of homogeneous polynomial equations over some field K. When are we guaranteed a (nontrivial) solution in K? Brauer famously proved that for many fields of interest (e.g. Q[i]), it is enough that the number of variables be sufficiently large. Birch proved that for the rational numbers (or any number field) this also holds, as long as the polynomials have odd degree. We will survey the history of this problem and present recent work improving its quantitative and qualitative aspects. A key tool we use comes from additive combinatorics. Partly based on joint work with Andrew Snowden and Tamar Ziegler.