Abstract: The hydrodynamic free boundary problem known as Laplacian Growth (or the Hele–Shaw problem) describes the evolution of a thin region of viscous fluid sandwiched between two flat plates. Regions between the plates not occupied by viscous fluid are assumed to be occupied by a much less viscous fluid like air. In this talk we consider modeling of Laplacian Growth making use of conformal maps of canonical domains. We give Hamiltonian and Lagrangian interpretations of Laplacian growth by an action functional. Its variation admits some interpretations in terms of Conformal Field Theory. In particular, it represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.