Given a ring $C$ and a totally (resp. partially) ordered
set of "monomials" $\mathfrak{M}$, Hahn (resp. Higman)
defined the set of power series $C [[ \mathfrak{M} ]]$ with
well-ordered (resp. Noetherian or well-quasi-ordered) support
in $\mathfrak{M}$. This set $C [[ \mathfrak{M} ]]$ can usually
be given a lot of additional structure: if $C$ is a field and
$\mathfrak{M}$ a totally ordered group, then Hahn proved that
$C [[ \mathfrak{M} ]]$ is a field. More recently, we have
constructed fields of ``transseries'' of the form $C [[
\mathfrak{M} ]]$ on which we defined natural derivations and
compositions. In this paper we develop an operator theory for
generalized power series of the above form. We first study
linear and multilinear operators. We next isolate a big class
of so-called Noetherian operators $\Phi : C [[ \mathfrak{M} ]]
\rightarrow C [[ \mathfrak{N}]]$, which include (when defined)
summation, multiplication, differentiation, composition,
etc. Our main result is the proof of an implicit function
theorem for Noetherian operators. This theorem may be used to
explicitly solve very general types of functional equations in
generalized power series.