I'll start with Bergman's 1971 definition of the logarithmic limit set of an ideal, and explain why we now call it the "tropicalization" of the ideal. I'll then explain how, when we study tropicalizations of linear spaces, we get the Bergmann complex of a matroid, which Yucong spoke about last week. Finally, I'll try to motivate the wonderful compactification of a hyperplane arrangement complement. Expect a fairly informal, loosely prepared, talk.