The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest Σ 3 ×SO(8) structure in this framework.