Abstract: Matroids are combinatorial abstractions of vector spaces embedded in a coordinate space.  Many fundamental questions have been open for these classical objects.  We highlight some recent progress that arise from the interaction between matroid theory and algebraic geometry.  Key objects involve compactifications of embedded vector spaces, and an exceptional Hirzebruch-Riemann-Roch isomorphism between the K-ring of vector bundles and the cohomology ring of stellahedral varieties.