  We consider the problem of constructing reduced models for large scale
systems with poles in general domains in the complex plane (as opposed to,
e.g., the open left-half plane or the open unit disk). Our goal is to design a
model reduction scheme, building upon theoretically established methodologies,
yet encompassing this new class of models. To this aim, we develop a balanced
truncation framework through conformal maps to handle poles in general domains.
The major difference from classical balanced truncation resides in the
formulation of the Gramians. We show that these new Gramians can still be
computed by solving modified Lyapunov equations for specific conformal maps. A
numerical algorithm to perform balanced truncation with conformal maps is
developed and is tested on three numerical examples, namely a heat model, the
Schr\"odinger equation, and the undamped linear wave equation, the latter two
having spectra on the imaginary axis.
