I shall begin by describing the real 8-dimensional division algebra O, discovered by Graves in 1843 and properly known as the octonions. With their use, a certain 27-dimensional real vector space H may easily be defined, which plays host to many fascinating geometrical and topological phenomena. These interact beautifully with the symmetries of H, which form one of the exceptional compact Lie groups so influentially described by Frank Adams in his posthumous book. Without giving too much of the game away, I hope to use this approach to introduce the audience to octonionic versions of projective spaces, flag manifolds, and Hopf maps. By way of conclusion I shall recall the octonionic Stiefel manifold question, which was posed by James in the 1950s and appears (to me, so far) to have lain around unanswered ever since.