  We study conformal maps from multiply connected domains in the extended
complex plane onto lemniscatic domains. Walsh proved the existence of such maps
in 1956 and thus obtained a direct generalization of the Riemann mapping
theorem to multiply connected domains. For polynomial pre-images of simply
connected sets we derive a construction principle for Walsh's conformal map in
terms of the Riemann map for the simply connected set. Moreover, we explicitly
construct examples of Walsh's conformal map for certain radial slit domains and
circular domains.
