  We construct a generalization of the theory of symmetric functions involving
functions of commuting and anticommuting (Grassmannian) variables. These new
functions, called symmetric functions in superspace, are invariant under the
diagonal action of the symmetric group acting on the sets of commuting and
anticommuting variables. We first obtain superspace analogues of a number of
standard objects and concepts in the theory of symmetric functions: partitions,
monomials, elementary symmetric functions, completely symmetric functions,
power sums, involutions, generating functions, Cauchy formulas, and scalar
products. We then consider a one-parameter extension of the combinatorial
scalar product. It provides the natural setting for the definition of a family
of ``combinatorial'' orthogonal Jack polynomials in superspace. We show that
this family coincides with that of ``physical'' Jack polynomials in superspace
that were previously introduced by the authors as orthogonal eigenfunctions of
a supersymmetric quantum mechanical many-body problem. The equivalence of the
two families is established by showing that the ``physical'' Jack polynomials
are also orthogonal with respect to the combinatorial scalar product. This
equivalence is also directly demonstrated for particular values of the free
parameter.
