  Extending our prior investigation, we give a new off-shell construction of
theories of spinning particles propagating in Minkowski spaces with arbitrary
$N$-extended supersymmetry on the world-line. The basis of the new off-shell
formulation is provided by realizations of new algebraic structures ${\cal
G}{\cal R}$(${\rm d}, N$) that are certain generalizations of Pauli algebras.
