Multiferroic hexagonal TbMnO3 thin films have been epitaxially stabilized on substrates with hexagonal in-plane symmetry by pulsed laser deposition. The hexagonal TbMnO3 films show ferroelectricity below ca. 60 K with a remnant polarization 20 times larger compared to its orthorhombic bulk phase. In addition, these samples show an electric-field-induced antiferroelectric-to-ferroelectric phase transition.