A study of the geometry of submanifolds of real 8-space under the group of motions generated by translations and rotations in the subgroup Spin(7) instead of the full SO(8). I call real 8-space endowed with this group O or octonian space. The fact that the stabilizer of an oriented 2-plane in Spin(7) is U(3) implies that any oriented 6-manifold in O inherits a U(3)-structure. The first part of the paper studies the generality of the 6-manifolds whose inherited U(3)-structure is symplectic, complex, or Kähler, etc. by applying the theory of exterior differential systems. I then turn to the study of the standard 6-sphere in O as an almost complex manifold and study the space of what are now called pseudo-holomorphic curves in the 6-sphere. I prove that every compact Riemann surface occurs as a (possibly ramified) pseudo-holomorphic curve in the 6-sphere. I also show that all of the genus zero pseudo-holomorphic curves in the 6-sphere are algebraic as surfaces. Reprints are available.