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The theory of matroids or combinatorial geometries originated in linearalgebra and graph theory, and has deep connections with many other areas,including field theory, matching theory, submodular optimization, Liecombinatorics, and total positivity. Matroids capture the combinatorial essencethat these different settings share. In recent years, the (classical,polyhedral, algebraic, and tropical) geometric roots of the field have grownmuch deeper, bearing new fruits. We survey some recent successes, stemming fromthree geometric models of a matroid: the matroid polytope, the Bergman fan, andthe conormal fan.

The geometry of geometries: matroid theory, old and new