[en] We present new spacetime algebras whose existence is due to special Γ-matrix identities which are also necessary for the existence of super p-branes. They contain a p-th rank antisymmetric tensor, and a (p-1) rank antisymmetric tensor-spinor generator. Furthermore the translations do not commute with the supercharge. In the case of supermembranes, we find a super-Kac-Moody extension of the new spacetime algebra. We give a realization of the algebra in terms of operators in d=11 superspace, and find a connection with the d=11 supermembrane action which leads to an elegant supergeometric formulation. By double dimensional reduction, we obtain a Kac-Moody algebra for the Type IIA Green-Schwarz superstring. We also discuss the generalization of these results, including the construction of generalized super-Virasoro algebras, for super p-branes. (author). 21 refs