In this chapter, we explore another generalization of C and IE, a non-associative real algebra, the Cayley algebra of octonions, (D. Like complex numbers and quaternions, octonions f o r m a real division algebra, of the highest possible dimension, 8. As an extreme case, @ makes its presence felt in classifications, for instance, in conjunction with exceptional cases of simple Lie algebras. Like C and lE, @ has a geometric interpretation. The automorphism group of lE is SO(3), the rotation group of IR 3 in IHI = IR| 3. The automorphism group of 9 = IR @ i~7 is not all of SO(7), but only a subgroup, the exceptional LŸ group G2. The subgroup G2 fixes a 3-vector, in A s IR T, whose choice determines the product rule of (D. The Cayley algebra @ is a tool to handle an esoteric phenomenon in dimension 8, namely triality, an automorphism of the universal covering group Spin(8) of the rotation group SO(8) of the Euclidean space E s. In general, all automorphisms of SO(n) ate either inner or similarities by orthogonal matrices in O(n), and all automorphisms of Spin(n) ate restrictions of linear transformations Cgn --+ Cgn, and project down to automorphisms of SO(n). The only exception is the triality automorphism of Spin(8), which cannot be linear while it permutes cyclically the three non-identity elements -1 , e12...s,-e12...8 in the center of Spin(8).