The notion of an anti-commutative (resp. commutative) rigid superalgebra is a
natural generalisation of the notion of a Lie (resp. Jordan) superalgebra.
Intuitively rigidity means that small deformations of the product under the
structural group produce an isomorphic algebra. In this paper we classify all
linearly compact simple anti-commutative (resp. commutative) rigid
superalgebras. Beyond Lie (resp. Jordan) superalgebras the complete list
includes four series and twenty two exceptional superalgebras (resp. ten
exceptional superalgebras).