I will discuss homogeneous convex cones. We first characterize the extreme rays of such cones in the context of their primal

construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results,

we prove that every homogeneous cone is facially exposed. We provide an alternative proof of a result of G\"uler and Tun\c{c}el that the Siegel rank of a symmetric cone is equal to its Carath\'eodory number. Our proof does not use the Jordan-vonNeumann-Wigner

characterization of the symmetric cones but it easily follows from the primal construction of the homogeneous cones and

our results on the geometry of homogeneous cones in primal and dual forms. We briefly discuss the duality mapping in the context

of automorphisms of convex cones and prove that the duality mapping is not an involution on certain self-dual cones.

This is based on joint work with Van Anh Truong.