Many properties of line bundles and vector bundles of higher rank become evident from the convex polytopes associated to their global sections. If the vector bundle has rank higher than one certain useful and nice equivalences are no longer valid. This talk will recall the dictionary between convex geometry and positivity in algebraic geometry, available for line bundles, and present the corresponding picture for higher rank toric vector bundles. The higher rank results are joint work with K. Jabbusch and G.G. Smith.