  The notion of "N = 2 vertex superalgebra with two odd formal variables" is
presented, the main axiom being a Jacobi identity with odd formal variables in
which an N=2 superconformal shift is incorporated into the usual Jacobi
identity for a vertex superalgebra. It is shown that as a consequence of these
axioms, the N=2 vertex superalgebra is naturally a representation of the Lie
algebra isomorphic to the three-dimensional algebra of superderivations with
basis consisting of the usual conformal operator and the two N=2 superconformal
operators. The notion of N=2 Neveu-Schwarz vertex operator superalgebra with
two odd formal variables is introduced, and consequences of this notion are
derived. Various other formulations of the notion of N=2 (Neveu-Schwarz) vertex
(operator) superalgebra appearing in the mathematics and physics literature are
discussed, and several mistakes in the literature are noted and corrected. The
notion of ``N=2 (Neveu-Schwarz) vertex (operator) superalgebra with one odd
formal variable" is formulated. It is shown that this formulation naturally
arises from alternate notions of N=1 superconformality and the continuous
deformation of an N=1 (Neveu-Schwarz) vertex (operator) superalgebra with one
odd formal variable. This notion is formulated to reflect the underlying N=1
superanalytic geometry, and it is shown that the equivalence of the notions of
N=2 (Neveu-Schwarz) vertex (operator) superalgebra with one and with two odd
formal variables reflects the equivalence of N=2 superconformal and N=1
superanalytic geometry. In particular we prove that the group of formal N=2
superconformal functions vanishing at zero and invertible in a neighborhood of
zero is isomorphic to a certain subgroup of N=1 superanalytic functions
vanishing at zero and invertible in a neighborhood of zero.
