  The complete classification of the irreducible representations of the
N-extended one-dimensional supersymmetry algebra linearly realized on a finite
number of fields is presented. Off-shell invariant actions of one-dimensional
supersymmetric sigma models are constructed. The role of both Clifford algebras
and the Cayley-Dickson's doublings of algebras in association with the
N-extended supersymmetries is discussed. We prove in specific examples that the
octonionic structure constants enter the N=8 invariant actions as coupling
constants. We further explain how to relate one-dimensional supersymmetric
quantum mechanical systems to the dimensional reduction of higher-dimensional
supersymmetric theories.
