Thirty years ago, Deligne gave a canonical increasing filtration on the singular cohomology of a complex algebraic variety, called the weight filtration. This filtration is respected by natural operations such as cup product and pullbacks, and is closely related to properties of compactifications and resolutions of singularities. I will discuss recent attempts, inspired by tropical and nonarchimedean analytic geometry, to understand this weight filtration as combinatorially as possible.