  We classify the convex polytopes whose symmetry groups have two orbits on the
flags. These exist only in two or three dimensions, and the only ones whose
combinatorial automorphism group is also two-orbit are the cuboctahedron, the
icosidodecahedron, and their duals. The combinatorially regular two-orbit
convex polytopes are certain 2n-gons for each n > 1. We also classify the
face-to-face tilings of Euclidean space by convex polytopes whose symmetry
groups have two flag orbits. There are finitely many families, tiling one, two,
or three dimensions. The only such tilings which are also combinatorially
two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the
tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb.
