  The theory of symmetric functions has been extended to the case where each
variable is paired with an anticommuting one. The resulting expressions, dubbed
superpolynomials, provide the natural N=1 supersymmetric version of the
classical bases of symmetric functions. Here we consider the case where two
independent anticommuting variables are attached to each ordinary variable. The
N=2 super-version of the monomial, elementary, homogeneous symmetric functions,
as well as the power sums, are then constructed systematically (using an
exterior-differential formalism for the multiplicative bases), these functions
being now indexed by a novel type of superpartitions. Moreover, the scalar
product of power sums turns out to have a natural N=2 generalization which
preserves the duality between the monomial and homogeneous bases. All these
results are then generalized to an arbitrary value of N. Finally, for N=2, the
scalar product and the homogenous functions are shown to have a one-parameter
deformation, a result that prepares the ground for the yet-to-be-defined N=2
Jack superpolynomials.
