The authors present the geometry of convex cones which are interesting for their potential in interior-point method theories and duality results for convex optimization problems. They characterize the extreme rays of homogeneous convex cones in primal and dual form and consequently it is shown that such a cone is facially exposed thereby generalizing some well-known results. The authors also study self-concordant barriers for homogeneous convex cones and it is proved that for self-dual cones that are not homogeneous, the duality mapping is not necessarily an involution. They point out that any conic optimization problem with a homogeneous cone as the cone constraint can be expressed as a semi-definite programming problem in principle.