  Huang's geometric interpretation of vertex operator algebras is extended to a
supergeometric interpretation of vertex operator superalgebras. In particular,
the geometry of spheres with punctures and local analytic coordinates in terms
of exponentials of derivations is extended to the geometry of superspheres with
punctures, a given spin structure, and local superconformal coordinates in
terms of exponentials of superderivations. The notion of supergeometric vertex
operator superalgebra is introduced, and in addition, the notion of
(superalgebraic) vertex operator superalgebra over a Grassmann algebra and with
odd formal variables is introduced. The main result is that the category of
supergeometric vertex operator superalgebras over a Grassmann algebra with a
given spin structure and the category of vertex operator superalgebras over a
Grassmann algebra are isomorphic. In addition, the supergeometric symmetry
between the two different possible spin structures naturally gives rise to a
symmetry in the superalgebraic structure.
