From the algebraic point of view, a multiple Fock space over ℌ is nothing but a standard (symmetric or antisymmetric) Fock space over a direct sum of copies of ℌ,and new definitions are not necessary in principle. However, this chapter contains important new notation, and a few rather interesting constructions, like that of the “finite-temperature” (= extremal universally invariant) representations of the CCR. In classical probability, it corresponds to the passage from one-dimensional to several-dimensional Brownian motion, necessary for Ito’s theory of stochastic differential equations.