  We consider Walsh's conformal map from the exterior of a compact set $E
\subseteq \mathbb{C}$ onto a lemniscatic domain. If $E$ is simply connected,
the lemniscatic domain is the exterior of a circle, while if $E$ has several
components, the lemniscatic domain is the exterior of a generalized lemniscate
and is determined by the logarithmic capacity of $E$ and by the exponents and
centers of the generalized lemniscate. For general $E$, we characterize the
exponents in terms of the Green's function of $E^c$. Under additional symmetry
conditions on $E$, we also locate the centers of the lemniscatic domain. For
polynomial pre-images $E = P^{-1}(\Omega)$ of a simply-connected infinite
compact set $\Omega$, we explicitly determine the exponents in the lemniscatic
domain and derive a set of equations to determine the centers of the
lemniscatic domain. Finally, we present several examples where we explicitly
obtain the exponents and centers of the lemniscatic domain, as well as the
conformal map.
