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The notion of "N = 2 vertex superalgebra with two odd formal variables" ispresented, the main axiom being a Jacobi identity with odd formal variables inwhich an N=2 superconformal shift is incorporated into the usual Jacobiidentity for a vertex superalgebra. It is shown that as a consequence of theseaxioms, the N=2 vertex superalgebra is naturally a representation of the Liealgebra isomorphic to the three-dimensional algebra of superderivations withbasis consisting of the usual conformal operator and the two N=2 superconformaloperators. The notion of N=2 Neveu-Schwarz vertex operator superalgebra withtwo odd formal variables is introduced, and consequences of this notion arederived. Various other formulations of the notion of N=2 (Neveu-Schwarz) vertex(operator) superalgebra appearing in the mathematics and physics literature arediscussed, and several mistakes in the literature are noted and corrected. Thenotion of ``N=2 (Neveu-Schwarz) vertex (operator) superalgebra with one oddformal variable" is formulated. It is shown that this formulation naturallyarises from alternate notions of N=1 superconformality and the continuousdeformation of an N=1 (Neveu-Schwarz) vertex (operator) superalgebra with oneodd formal variable. This notion is formulated to reflect the underlying N=1superanalytic geometry, and it is shown that the equivalence of the notions ofN=2 (Neveu-Schwarz) vertex (operator) superalgebra with one and with two oddformal variables reflects the equivalence of N=2 superconformal and N=1superanalytic geometry. In particular we prove that the group of formal N=2superconformal functions vanishing at zero and invertible in a neighborhood ofzero is isomorphic to a certain subgroup of N=1 superanalytic functionsvanishing at zero and invertible in a neighborhood of zero.

On axiomatic aspects of N=2 vertex superalgebras with odd formal variables, and deformations of N=1 vertex superalgebras