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The algebraic structure on the subspace of the quasi-primary vectors given by the projection of the (n) products of a conformal superalgebra is formulated. As an application the complete list of simple physical conformal superalgebras is given. The list contains a one-parameter family of superconformal algebras with 4 supercharges that is simple for general values. * Email: yamamo@ms.u-tokyo.ac.jp PreliminariesLet K be a subfield ofThe homomorphisms of Z/2Z-graded vector spaces are supposed to be compatible with the gradation. The Z/2Z-gradation is called parity. V 0 is called the subspace of even parity, and V 1 is of odd parity. The Z/2Z-graded objects are called super-objects. Commutativity for the product · of a superalgebra is defined to be a · b = (−1) p(a)p(b) b · a, where a, b are supposed to be homogeneous with respect to the parity p. Now let us state the axioms for conformal superalgebras, based on the descriptions in [7] and [8]. We denote A (j) = A j /j!, where A is an operator.Definition 2.1 Let R be a Z/2Z-graded K-vector space equipped with countably many products (n) : R ⊗ R → R, (n ∈ N),

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