The Grassmannian parametrizes linear spaces in an n-dimensional vector space. In the same way, the tropicalization of the Grassmannian parameterizes topical linear space in R^n. The combinatorics describing tropical linear spaces leads to studying subdivisions of matroid polytopes into smaller matroid polytopes. This in turn reveals surprising new invariants of matroids.

This talk will attempt to survey work by many authors, including Kapranov; Ardila; Ardila and Klivans; Fink and Derksen; Hacking, Keel and Tevelev as well as the my own work together with Sturmfels and with Fink.