Motivated by studying the amoeba of a system of polynomial equations, we associate to each matroid $M$ a polyhedral complex $B(M)$, called the ``Bergman complex". I will describe the topology and combinatorics of this complex. Somewhat surprisingly, the space of phylogenetic trees is (essentially) a Bergman complex, and we obtain some new results about it as a consequence. If $M$ is oriented, the Bergman complex $B(M)$ has a ``positive part" $B+(M)$, which I will also describe. \vskip .1in \noindent If time allows, I will show that for a Coxeter arrangement $A$, $B(A)$ is closely related to de Concini and Procesi's compactification of the complement of $A$, and $B+(A)$ is dual to a known Coxeter generalization of the associahedron. \vskip .1in \noindent My talk will assume no previous knowledge of matroids and arrangements. \vskip .1in \noindent Parts of this work are joint with Carly Klivans, Lauren Williams, and Vic Reiner.